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b(Therefore,)29 b(it)f(is)f(necessary)i(to)g(describ)s(e)e(the)i(k)m(ey)g(features)f (of)1854 5652 y(3)p eop %%Page: 4 4 4 3 bop 382 548 a Fp(the)31 b(resolving)e(algorithm)g(in)g(more)i (details.)40 b(Besides,)30 b(w)m(e)h(ha)m(v)m(e)h(used)d(a)i(p)s (ossibilit)m(y)382 661 y(to)i(decrease)h(the)e(n)m(um)m(b)s(er)f(of)i (so-called)f(v)m(ertex-saturated)j(digraphs)30 b(that)j(ha)m(v)m(e)h (to)382 774 y(b)s(e)28 b(constructed)h(during)d(searc)m(hing)i(for)h(a) g(solution)e(of)h(MCPP)-8 b(.)29 b(It)g(has)f(en)m(tailed)g(the)382 887 y(necessit)m(y)j(to)g(mo)s(dify)d(the)j(initial)d(algorithm)h (appropriately)-8 b(.)382 1173 y Fl(2)135 b(P)l(artially)47 b(Ordered)e(Sets)382 1376 y Fp(Relations)35 b(b)s(et)m(w)m(een)i(t)m(w) m(o)h(ob)5 b(jects)37 b(are)f(called)g Fg(binary)p Fp(.)58 b(A)37 b(binary)d(relation)h Fk(R)i Fp(can)382 1489 y(b)s(e)30 b(represen)m(ted)g(b)m(y)g(a)h(list)e(of)i(all)e(pairs,)g(whic)m(h)g (are)i(in)e(the)i(relation)e Fk(R)q Fp(:)1458 1693 y(\()p Fk(a)1541 1707 y Fh(1)1581 1693 y Fk(;)15 b(b)1660 1707 y Fh(1)1700 1693 y Fp(\))p Fk(;)g(:)g(:)g(:)i(;)e Fp(\()p Fk(a)2020 1707 y Fj(m)2088 1693 y Fk(;)g(b)2167 1707 y Fj(m)2234 1693 y Fp(\))p Fk(:)382 1897 y Fp(If)35 b(\()p Fk(a;)15 b(b)p Fp(\))36 b Fi(2)e Fk(R)j Fp(then)e(w)m(e)i(designate)f (it)f(as)h Fk(aR)q(b)p Fp(.)57 b(When)36 b(\()p Fk(a;)15 b(b)p Fp(\))36 b Fi(62)e Fk(R)q Fp(,)j(w)m(e)f(will)d(write)382 2010 y Fk(a)450 1987 y Fp(\026)430 2010 y Fk(R)q(b)p Fp(.)51 b(If)33 b Fk(a)e Fi(2)g Fk(A)j Fp(and)f Fk(b)e Fi(2)g Fk(A)i Fp(for)h(all)f Fk(aR)q(b)g Fp(then)h Fk(R)g Fp(is)f(called)g(a)h Fg(r)-5 b(elation)38 b(on)e(the)h(set)382 2123 y Fk(A)p Fp(.)k(F)-8 b(urther)30 b(w)m(e)h(consider)e(only)g (relations)h(on)g(a)h Fg(\014nite)f Fp(set)h Fk(A)p Fp(.)523 2236 y(If)38 b Fk(A)g Fp(is)f(a)h(\014nite)f(set)i(and)e Fk(R)i Fp(is)e(a)h(relation)f(on)h Fk(A)p Fp(,)j(w)m(e)d(can)g (represen)m(t)g Fk(R)h Fp(as)f(a)382 2349 y(digraph)729 2326 y Fk(~)716 2349 y(G)o Fp(.)k(Eac)m(h)31 b(elemen)m(t)g(of)g Fk(A)g Fp(is)f(assigned)g(to)h(a)g(v)m(ertex)h(of)2652 2326 y Fk(~)2638 2349 y(G)o Fp(,)f(and)g(a)g(v)m(ertex)g Fk(a)3342 2363 y Fj(i)382 2462 y Fp(is)e(joined)h(with)f(a)h(v)m(ertex) i Fk(a)1350 2476 y Fj(j)1417 2462 y Fp(b)m(y)e(the)h(arc)f(\()p Fk(a)1934 2476 y Fj(i)1963 2462 y Fk(;)15 b(a)2051 2476 y Fj(j)2088 2462 y Fp(\))31 b(if)e(and)h(only)f(if)g Fk(a)2744 2476 y Fj(i)2773 2462 y Fk(R)q(a)2891 2476 y Fj(j)2927 2462 y Fp(.)523 2575 y(A)23 b(relation)g Fk(R)g Fp(is)g Fg(r)-5 b(e\015exive)23 b Fp(if)f Fk(aR)q(a)h Fp(for)g(ev)m(ery)h Fk(a)h Fi(2)g Fk(A)p Fp(.)38 b(A)24 b(relation)e Fk(R)i Fp(is)e Fg(irr)-5 b(e\015exive)382 2688 y Fp(if)30 b Fk(a)534 2665 y Fp(\026)514 2688 y Fk(R)q(a)h Fp(for)f(ev)m(ery)i Fk(a)26 b Fi(2)g Fk(A)p Fp(.)43 b(A)31 b(relation)f Fk(R)i Fp(is)e Fg(symmetric)i Fp(if)e Fk(bR)q(a)g Fp(whenev)m(er)h Fk(aR)q(b)p Fp(.)43 b(A)382 2801 y(relation)33 b Fk(R)h Fp(is)f Fg(antisymmetric)i Fp(if)e Fk(a)e Fp(=)g Fk(b)i Fp(whenev)m(er)h Fk(aR)q(b)f Fp(and)h Fk(bR)q(a)p Fp(.)50 b(A)34 b(relation)f Fk(R)382 2914 y Fp(is)c Fg(tr)-5 b(ansitive)32 b Fp(if)d Fk(aR)q(c)i Fp(whenev)m(er)f Fk(aR)q(b)g Fp(and)g Fk(bR)q(c)p Fp(.)523 3027 y(A)35 b(binary)f(relation)g Fk(R)i Fp(is)f(called)f(a)i Fg(p)-5 b(artial)39 b(or)-5 b(der)37 b Fp(if)d Fk(R)i Fp(is)f(an)m(tisymmetric)f(and)382 3139 y(transitiv)m(e.)65 b(A)39 b(set)g Fk(A)g Fp(together)h(with)e(a)h(partial)e(order)i Fk(R)g Fp(are)g(called)f(a)h Fg(p)-5 b(artial)5 b(ly)382 3252 y(or)-5 b(der)g(e)g(d)38 b(set)p Fp(.)50 b(W)-8 b(e)34 b(designate)f(this)g(partially)e(ordered)i(set)h(as)f(\()p Fk(A;)15 b(R)q Fp(\))34 b(or)g(\()p Fk(A;)15 b Fi(\025)p Fp(\).)50 b(If)382 3365 y(the)32 b(relation)f Fk(R)i Fp(is)e(irre\015exiv)m(e)g(then)h(its)f(partial)g(order)h(is)f(called)g Fg(strict)p Fp(.)46 b(A)33 b(strictly)382 3478 y(ordered)d(set)h(is)e (designated)h(as)h(\()p Fk(A;)15 b(>)p Fp(\).)523 3591 y(In)34 b(Fig.)52 b(1)35 b(\(a\))h(the)e(acyclic)h(graph)e(represen)m (ts)i(the)f(partial)f(order)h(induced)f(b)m(y)382 3704 y(the)d(binary)f(relation)927 3908 y Fk(R)d Fp(=)f Fi(f)p Fp(\()p Fk(a)1246 3922 y Fh(1)1286 3908 y Fk(;)15 b(a)1374 3922 y Fh(2)1414 3908 y Fp(\))p Fk(;)g Fp(\()p Fk(a)1572 3922 y Fh(1)1612 3908 y Fk(;)g(a)1700 3922 y Fh(3)1740 3908 y Fp(\))p Fk(;)g Fp(\()p Fk(a)1898 3922 y Fh(1)1939 3908 y Fk(;)g(a)2027 3922 y Fh(5)2067 3908 y Fp(\))p Fk(;)g Fp(\()p Fk(a)2225 3922 y Fh(4)2265 3908 y Fk(;)g(a)2353 3922 y Fh(2)2393 3908 y Fp(\))p Fk(;)g Fp(\()p Fk(a)2551 3922 y Fh(5)2592 3908 y Fk(;)g(a)2680 3922 y Fh(2)2720 3908 y Fp(\))p Fi(g)p Fk(:)382 4112 y Fp(Here)31 b(and)f(throughout)g (w)m(e)h(assume)f(that)h(the)g(orien)m(tation)f(of)h(digraph)d(arcs)j (repre-)382 4225 y(sen)m(ted)g(in)e(\014gures)g(is)h(from)g(b)s(elo)m (w)f(up)m(w)m(ards.)523 4338 y(Let)i(there)f(b)s(e)g(a)h(partially)d (ordered)i(set)h(\()p Fk(A;)15 b Fi(\025)p Fp(\).)523 4451 y(If)32 b Fk(a)c Fi(\025)g Fk(b)k Fp(or)h Fk(b)28 b Fi(\025)g Fk(a)k Fp(then)g(the)h(elemen)m(ts)f Fk(a)h Fp(and)e Fk(b)h Fp(of)h Fk(A)f Fp(are)h(called)e Fg(c)-5 b(omp)g(ar)g(able)p Fp(.)382 4564 y(If)30 b Fk(a)25 b Fi(6\025)g Fk(b)30 b Fp(and)g Fk(b)25 b Fi(6\025)g Fk(a)31 b Fp(then)f(suc)m(h)g(a)h(pair)e(of)h(elemen)m(ts)h(is)e(called)h Fg(inc)-5 b(omp)g(ar)g(able)p Fp(.)523 4677 y(If)36 b Fk(A)688 4691 y Fh(1)763 4677 y Fi(\022)f Fk(A)i Fp(and)f(eac)m(h)i (pair)d(of)h(elemen)m(ts)h(of)g Fk(A)2226 4691 y Fh(1)2302 4677 y Fp(is)f(comparable)g(then)g(w)m(e)h(sa)m(y)382 4790 y(that)e Fk(A)651 4804 y Fh(1)725 4790 y Fp(determines)e(a)i Fg(p)-5 b(ath)36 b Fp(of)e(\()p Fk(A;)15 b Fi(\025)p Fp(\).)54 b(If)33 b Fk(A)2065 4804 y Fh(1)2137 4790 y Fi(\022)e Fk(A)k Fp(and)e(eac)m(h)j(pair)d(of)h(elemen)m(ts)382 4903 y(of)40 b Fk(A)563 4917 y Fh(1)643 4903 y Fp(is)f(incomparable)g (then)g(w)m(e)i(sa)m(y)g(that)f Fk(A)2116 4917 y Fh(1)2196 4903 y Fp(is)f(an)h Fg(antip)-5 b(ath)43 b Fp(of)d(\()p Fk(A;)15 b Fi(\025)p Fp(\).)71 b(An)382 5016 y(an)m(tipath)33 b Fk(A)819 5030 y Fh(1)892 5016 y Fp(is)f(called)g Fg(maximum)j Fp(in)d(\()p Fk(A;)15 b Fi(\025)p Fp(\))34 b(if)e Fk(C)7 b(ar)s(d)p Fp(\()p Fk(A)2460 5030 y Fh(1)2499 5016 y Fp(\))31 b Fi(\025)e Fk(C)7 b(ar)s(d)p Fp(\()p Fk(A)2979 4983 y Fv(\003)3019 5016 y Fp(\))33 b(for)g(an)m(y)382 5129 y(an)m(tipath)d Fk(A)816 5096 y Fv(\003)881 5129 y Fi(\022)25 b Fk(A)30 b Fp(in)f(\()p Fk(A;)15 b Fi(\025)p Fp(\).)1854 5652 y(4)p eop %%Page: 5 5 5 4 bop 523 548 a Fp(W)-8 b(e)33 b(sa)m(y)g(that)f(a)g(partially)e (ordered)h(set)i(\()p Fk(A;)15 b Fi(\025)p Fp(\))32 b(is)f(partitioned) g(in)m(to)g(paths)h Fk(A)3306 562 y Fh(1)3345 548 y Fp(,)382 661 y(.)15 b(.)h(.)f(,)31 b Fk(A)627 675 y Fj(m)724 661 y Fp(if)e(eac)m(h)j Fk(A)1080 675 y Fj(i)1138 661 y Fp(\()p Fk(A)1241 675 y Fj(i)1295 661 y Fi(6)p Fp(=)25 b Fi(\013)p Fp(,)30 b Fk(i)c Fp(=)p 1670 592 166 4 v 25 w(1)p Fk(;)15 b(m)q Fp(\))30 b(is)g(a)g(path,)1682 820 y Fj(m)1667 845 y Ff([)1656 1027 y Fj(i)p Fh(=1)1785 926 y Fk(A)1853 940 y Fj(i)1907 926 y Fp(=)25 b Fk(A;)382 1203 y Fp(and)30 b Fk(A)627 1217 y Fj(i)675 1203 y Fi(\\)20 b Fk(A)824 1217 y Fj(j)886 1203 y Fp(=)25 b Fi(\013)30 b Fp(whenev)m(er)g Fk(i)c Fi(6)p Fp(=)f Fk(j)36 b Fp(\()p Fk(i;)15 b(j)32 b Fi(2)25 b(f)p Fp(1)p Fk(;)15 b(:)g(:)g(:)i(;)e(m)p Fi(g)p Fp(\).)523 1316 y(A)25 b(partition)e(of)i(\()p Fk(A;)15 b Fi(\025)p Fp(\))26 b(in)m(to)f(paths)f(is)g(called)g Fg(minimum)h Fp(if)f(it)g(has)g(the)h(minim)m(um)382 1429 y(n)m(um)m(b)s(er)j(of)i(elemen)m(t)g Fk(m)g Fp(comparing)f(with)f (other)i(partitions)f(of)g(\()p Fk(A;)15 b Fi(\025)p Fp(\))31 b(in)m(to)f(paths.)382 1542 y(Suc)m(h)c(a)h(partition)e(is)h (also)h(called)f(the)g Fg(minimum)k(p)-5 b(ath)31 b(p)-5 b(artition)32 b(\(MPP\))26 b Fp(of)g(\()p Fk(A;)15 b Fi(\025)p Fp(\).)523 1655 y(There)25 b(exists)g(famous)g(Dilw)m(orth's) f(theorem)i(that)g(establishes)e(relationship)f(b)s(e-)382 1768 y(t)m(w)m(een)31 b(an)g(MPP)f(and)g(the)g(maxim)m(um)g(an)m (tipath)g(of)g(\()p Fk(A;)15 b Fi(\025)p Fp(\))32 b([2,)f(9].)382 1980 y Fq(Theorem)j(1)46 b Fp(\(Dilw)m(orth)20 b(R.P)-8 b(.\))25 b Fg(L)-5 b(et)24 b Fp(\()p Fk(A;)15 b Fi(\025)p Fp(\))25 b Fg(b)-5 b(e)23 b(a)i(\014nite)f(p)-5 b(artial)5 b(ly)26 b(or)-5 b(der)g(e)g(d)26 b(set.)39 b(The)382 2093 y(minimum)c(numb)-5 b(er)35 b(of)g(disjoint)h(p)-5 b(aths,)37 b(which)e(the)h(set)e Fp(\()p Fk(A;)15 b Fi(\025)p Fp(\))36 b Fg(c)-5 b(an)35 b(b)-5 b(e)35 b(p)-5 b(artitione)g(d)382 2206 y(into,)33 b(e)-5 b(quals)33 b(to)g(the)g(c)-5 b(ap)g(acity)35 b(of)e(the)g(maximum)h(antip)-5 b(ath)35 b(in)d Fp(\()p Fk(A;)15 b Fi(\025)p Fp(\))p Fg(.)523 2419 y Fp(There)31 b(is)f(an)i(e\016cien)m(t)g(algorithm)e(to)i(partition)f(a)g(\014nite)g (partially)e(ordered)i(set)382 2532 y(in)m(to)44 b(the)g(minim)m(um)d (n)m(um)m(b)s(er)h(of)i(paths)f(and)g(to)i(\014nd)d(the)i(maxim)m(um)f (an)m(tipath,)382 2645 y(elab)s(orated)j(b)m(y)g(F)-8 b(ord)46 b(L.R.)g(and)g(F)-8 b(ulk)m(erson)46 b(D.R.)h([2].)88 b(In)46 b(essence,)51 b(this)45 b(algo-)382 2758 y(rithm)36 b(\014nds)f(the)j(maxim)m(um)e(matc)m(hing)i(in)e(a)i(bipartite)e (graph)h Fk(G)2779 2725 y Fv(\003)2855 2758 y Fp(=)f(\()p Fk(X)r(;)15 b(Y)5 b(;)15 b Fp(\000)3269 2725 y Fv(\003)3310 2758 y Fp(\).)382 2870 y(If)37 b(a)g(partially)f(set)i(has)f Fk(n)g Fp(elemen)m(ts)g(then)g(this)g(graph)f(con)m(tains)i(2)p Fk(n)f Fp(v)m(ertices)h(and)382 2983 y Fk(C)7 b(ar)s(d)p Fp(\()p Fk(X)g Fp(\))32 b(=)f Fk(C)7 b(ar)s(d)p Fp(\()p Fk(Y)19 b Fp(\))32 b(=)f Fk(n)p Fp(.)51 b(An)34 b(edge)g(\()p Fk(x)1947 2997 y Fj(i)1976 2983 y Fk(;)15 b(y)2061 2997 y Fj(j)2097 2983 y Fp(\))35 b(joins)d(t)m(w)m(o)k(v)m(ertices)e Fk(x)2945 2997 y Fh(1)3016 2983 y Fi(2)d Fk(X)41 b Fp(and)382 3096 y Fk(y)427 3110 y Fj(j)498 3096 y Fi(2)35 b Fk(Y)57 b Fp(if)35 b(and)h(only)g(if)f(the)i(corresp)s(onding)d(elemen)m(ts)j Fk(a)2450 3110 y Fh(1)2489 3096 y Fk(;)15 b(a)2577 3110 y Fh(2)2653 3096 y Fi(2)35 b Fk(A)h Fp(of)h(\()p Fk(A;)15 b Fi(\025)p Fp(\))37 b(are)382 3209 y(comparable.)523 3322 y(In)31 b(man)m(ual)f(computations,)i(w)m(e)g(will)d(use)i(the)g (adjacen)m(t)i(matrix)d Fk(M)42 b Fp(of)31 b Fk(G)3142 3289 y Fv(\003)3213 3322 y Fp(as)g(a)382 3435 y(w)m(orking)23 b(table.)39 b(Units)23 b(of)i Fk(M)34 b Fp(determine)23 b(its)h Fg(admissible)k(c)-5 b(el)5 b(ls)p Fp(.)39 b(Tw)m(o)24 b(cells)g(of)g Fk(M)34 b Fp(are)382 3548 y(called)21 b Fg(indep)-5 b(endent)23 b Fp(if)e(they)h(are)g(lo)s(cated)g(in)e (distinct)g(ro)m(ws)i(and)f(distinct)f(columns)g(of)382 3661 y Fk(M)10 b Fp(.)40 b(T)-8 b(o)30 b(\014nd)e(the)i(maxim)m(um)e (matc)m(hing,)i(w)m(e)g(ha)m(v)m(e)g(to)g(\014nd)e(the)i(maxim)m(um)e (n)m(um)m(b)s(er)382 3774 y(of)i(admissible)e(indep)s(enden)m(t)g (cells)h(of)i Fk(M)10 b Fp(.)523 3887 y(The)29 b(algorithm)g(for)h (partition)f(of)h(a)g(partially)e(ordered)h(set)h(in)m(to)g(the)g (minim)m(um)382 4000 y(n)m(um)m(b)s(er)f(of)h(paths)g(consists)g(of)h (t)m(w)m(o)h(stages:)518 4187 y Fi(\017)46 b Fp(Construct)30 b(an)g(initial)e(partition)h(of)i(the)f(partially)f(ordered)g(set)i(in) m(to)g(paths;)518 4375 y Fi(\017)46 b Fp(Impro)m(v)m(e)31 b(the)g(existing)e(partition)g(while)f(it)i(is)g(p)s(ossible.)523 4562 y(T)-8 b(o)42 b(a)m(v)m(oid)f(am)m(biguit)m(y)-8 b(,)44 b(w)m(e)e(alw)m(a)m(ys)g(lo)s(ok)e(through)h(ro)m(ws)g(and)f (columns)g(of)i Fk(M)382 4675 y Fp(uniformly:)c(from)30 b(top)g(to)h(b)s(ottom)g(in)e(columns)g(and)h(from)g(left)g(to)h(righ)m (t)f(in)f(ro)m(ws.)523 4788 y(T)-8 b(o)31 b(obtain)f(the)g(initial)e (partition,)h(w)m(e)i(ma)m(y)g(use)f(the)h(follo)m(wing)d(pro)s (cedure.)382 4973 y Fg(Step)33 b(0.)46 b Fp(Put)30 b Fk(N)35 b Fp(=)25 b Fk(n)p Fp(,)30 b(where)g Fk(n)g Fp(is)f(the)i(n)m (um)m(b)s(er)e(of)h(ro)m(ws)h Fk(M)10 b Fp(,)30 b Fk(i)c Fp(=)f(1.)382 5110 y Fg(Step)33 b(1.)46 b Fp(If)30 b Fk(N)35 b Fp(=)25 b(0,)31 b(complete)g(the)f(calculations)g(as)g(an)h (initial)c(partition)i(is)h(found.)382 5246 y Fg(Step)j(2.)46 b Fp(In)40 b(the)i Fk(i)p Fp(-th)f(ro)m(w)h(of)f Fk(M)10 b Fp(,)44 b(\014nd)c(the)h(\014rst)g(on)g(the)g(order)g(admissible)d (cell,)609 5359 y(whose)33 b(appropriate)f(column)g(is)g(not)h(mark)m (ed.)49 b(If)32 b(suc)m(h)h(a)g(cell)f(is)g(not)i(found,)1854 5652 y(5)p eop %%Page: 6 6 6 5 bop 609 548 a Fp(put)26 b Fk(i)g Fp(:=)f Fk(i)13 b Fp(+)g(1,)29 b Fk(N)35 b Fp(:=)25 b Fk(N)e Fi(\000)13 b Fp(1)28 b(and)e(go)i(to)f(Step)g(1.)40 b(Otherwise,)26 b(remem)m(b)s(er)g(the)609 661 y(found)k(cell)h(\()p Fk(i;)15 b(j)5 b Fp(\),)34 b(mark)d(a)g(column)f Fk(j)5 b Fp(,)33 b(calculate)f Fk(i)27 b Fp(:=)g Fk(i)21 b Fp(+)f(1)32 b(and)f(go)h(to)g(Step)609 774 y(1.)523 952 y(Fig.)49 b(1)33 b(\(b\))g(sho)m(ws)g(the)g(adjacen)m(t)i(matrix)d(for)h(the)g (strictly)f(ordered)g(set)i(repre-)382 1065 y(sen)m(ted)g(in)e(Fig.)51 b(1)34 b(\(a\).)52 b(The)33 b(c)m(hosen)h(cells)f(of)h(the)g(initial)d (partition)h(are)i(indicated)382 1178 y(b)m(y)c(circles.)523 1290 y(T)-8 b(o)36 b(\014nd)d(an)j(MPP)f(and)f(the)i(maxim)m(um)e(an)m (tipath)h(of)g(a)h(set)g(\()p Fk(A;)15 b Fi(\025)p Fp(\),)37 b(w)m(e)f(apply)382 1403 y(F)-8 b(ord-F)g(ulk)m(erson)35 b(algorithm)e([2)q(].)53 b(The)34 b(algorithm)f(runs)g(after)i(the)f (termination)f(of)382 1516 y(the)d(previous)e(pro)s(cedure,)h(that)h (is,)f(when)f(there)i(exists)f(an)h(initial)c(partition)j(of)g(the)382 1629 y(ordered)h(set)h(in)m(to)f(paths.)382 1807 y Fg(Step)j(1.)46 b Fp(Mark)26 b(ro)m(ws)g(of)g Fk(M)10 b Fp(,)27 b(not)f(con)m(taining)f (the)h(c)m(hosen)h(cells,)f(b)m(y)g(the)g(sym)m(b)s(ol)e(\()p Fi(\003)p Fp(\).)382 1944 y Fg(Step)33 b(2.)46 b Fp(Lo)s(ok)26 b(through)f(newly)f(mark)m(ed)i(ro)m(ws)g(of)g Fk(M)36 b Fp(and)25 b(\014nd)f(all)g(non-c)m(hosen)j(cells)609 2057 y(in)f(eac)m(h)i(ro)m(w.)40 b(Mark)27 b(all)f(unmark)m(ed)g (columns)g(of)h Fk(M)10 b Fp(,)28 b(corresp)s(onding)d(to)j(suc)m(h)609 2170 y(cells,)i(b)m(y)g(index)f(of)i(the)f(ro)m(w.)382 2306 y Fg(Step)j(3.)46 b Fp(Lo)s(ok)32 b(through)e(newly)h(mark)m(ed)g (columns.)43 b(If)31 b(an)h(examined)f(column)f(con-)609 2419 y(tains)i(a)g(c)m(hosen)g(cell)g(\(that)h(is,)e(a)i(cell)e (enclosed)h(within)d(a)k(circle\))e(then)h(mark)609 2532 y(the)d(ro)m(w,)g(con)m(taining)f(the)g(c)m(hosen)h(cell,)f(b)m(y)g (index)f(of)i(the)f(examined)g(column.)609 2645 y(If)i(the)h (considered)e(column)g(do)s(es)h(not)h(con)m(tain)f(a)h(c)m(hosen)g (cell,)f(go)h(to)g(Step)f(4.)609 2758 y(If)g(it)g(is)f(imp)s(ossible)e (to)k(mark)g(new)e(ro)m(ws,)i(go)g(to)g(Step)f(5.)382 2894 y Fg(Step)j(4.)46 b Fp(The)25 b(essence)h(of)g(this)e(step)h(is)g (a)h(pro)s(cedure)e(constructing)h(a)h(new)f(collection)609 3007 y(of)32 b(indep)s(enden)m(t)d(cells,)j(ha)m(ving)f(one)h(more)g (cell)e(than)i(the)g(former)f(collection.)609 3120 y(A)m(t)38 b(eac)m(h)h(stage)f(of)g(this)e(pro)s(cedure,)i(except)g(for)f(the)h (\014nal)e(step,)j(w)m(e)e(pic)m(k)g(a)609 3233 y Fg(new)j Fp(admissible)d(cell)i(of)g Fk(M)50 b Fp(and)39 b(delete)h(an)f(\\old") h(one.)69 b(The)39 b(increase)g(of)609 3346 y(the)29 b(total)g(n)m(um)m(b)s(er)e(of)i(c)m(hosen)g(cells)f(happ)s(ens)e(as)j (follo)m(ws.)39 b(In)28 b(the)h(found)e Fk(j)5 b Fp(-th)609 3459 y(column,)35 b(c)m(ho)s(ose)g(a)g(new)f(cell)f(in)g(a)i(ro)m(w)f Fk(m)p Fp(\()p Fk(j)5 b Fp(\),)37 b(where)d Fk(m)p Fp(\()p Fk(j)5 b Fp(\))35 b(is)f(the)g(mark)g(of)609 3572 y(the)f(curren)m(t)f (column.)46 b(Let)33 b(w)m(e)g(ha)m(v)m(e)h(already)e(c)m(hosen)h(the)g (cell)f(\()p Fk(i;)15 b(j)5 b Fp(\),)35 b(whic)m(h)609 3685 y(the)26 b(marks)e Fk(m)p Fp(\()p Fk(i)p Fp(\))i(and)f Fk(m)p Fp(\()p Fk(j)5 b Fp(\))26 b(corresp)s(ond)e(to,)k(where)c Fk(m)p Fp(\()p Fk(i)p Fp(\))i(is)f(the)g(mark)g(of)g(the)609 3798 y Fk(i)p Fp(-th)30 b(ro)m(w.)40 b(If)29 b Fk(m)p Fp(\()p Fk(j)5 b Fp(\))26 b(=)f(\()p Fi(\003)p Fp(\),)31 b(then)e(the)g(construction)g(of)h(the)f(new)g(collection)f(of)609 3910 y(indep)s(enden)m(t)33 b(cells)h(is)f(accomplished.)53 b(Delete)36 b(all)d(ro)m(w)i(and)f(column)g(marks)609 4023 y(and)g(go)g(to)h(Step)f(2.)52 b(Otherwise,)34 b(delete)g(the)g (cell)g(\()p Fk(i;)15 b(m)p Fp(\()p Fk(i)p Fp(\)\))36 b(and)d(c)m(ho)s(ose)i(the)609 4136 y(cell)42 b(\()p Fk(m)p Fp(\()p Fk(m)p Fp(\()p Fk(i)p Fp(\)\),)48 b Fk(m)p Fp(\()p Fk(i)p Fp(\)\).)80 b(Put)42 b Fk(i)47 b Fp(=)f Fk(m)p Fp(\()p Fk(m)p Fp(\()p Fk(i)p Fp(\)\),)i Fk(j)j Fp(=)46 b Fk(m)p Fp(\()p Fk(i)p Fp(\))e(and)e(rep)s(eat)h(the)609 4249 y(describ)s(ed)28 b(ab)s(o)m(v)m(e)k(pro)s(cess.)382 4386 y Fg(Step)h(5.)46 b Fp(Find)37 b(the)i(maxim)m(um)e(an)m(tipath)i Fi(U)47 b Fp(=)38 b Fk(U)2175 4400 y Fj(r)2239 4386 y Fi(n)26 b Fk(U)2372 4400 y Fj(c)2407 4386 y Fp(,)41 b(where)d Fk(U)2806 4400 y Fj(r)2883 4386 y Fp(is)f(the)i(set)g(of)609 4499 y(mark)m(ed)c(ro)m(ws,)h(and)e Fk(U)1421 4513 y Fj(c)1490 4499 y Fp(is)f(the)i(set)g(of)g(mark)m(ed)f(columns.)52 b(T)-8 b(erminate.)53 b(The)609 4612 y(found)29 b(admissible)f(cells)h (determine)h(arcs)g(forming)f(the)i(paths)f(of)g(the)h(MPP)-8 b(.)523 4790 y(Fig.)50 b(1)34 b(\(c\))g(sho)m(ws)f(the)h(c)m(hosen)g (cells)e(of)i(the)g(optim)m(um)e(partition)g(for)h(the)h(par-)382 4902 y(tially)29 b(ordered)h(set,)i(represen)m(ted)f(in)f(Fig.)42 b(1)31 b(\(a\).)43 b(In)30 b(this)g(case,)i(the)f(MPP)g(consists)382 5015 y(of)d(the)f(paths)h Fk(A)950 5029 y Fh(1)1014 5015 y Fp(=)d Fi(f)p Fk(a)1203 5029 y Fh(1)1243 5015 y Fk(;)15 b(a)1331 5029 y Fh(3)1371 5015 y Fi(g)p Fp(,)29 b Fk(A)1538 5029 y Fh(2)1603 5015 y Fp(=)c Fi(f)p Fk(a)1792 5029 y Fh(4)1832 5015 y Fk(;)15 b(a)1920 5029 y Fh(2)1960 5015 y Fi(g)p Fp(,)28 b(and)f Fk(A)2300 5029 y Fh(3)2365 5015 y Fp(=)e Fi(f)p Fk(a)2554 5029 y Fh(5)2594 5015 y Fi(g)p Fp(.)40 b(W)-8 b(e)29 b(also)f(ha)m(v)m(e)g(the)382 5128 y(set)35 b Fk(U)590 5142 y Fj(r)660 5128 y Fp(=)d Fi(f)p Fk(a)856 5142 y Fh(2)896 5128 y Fk(;)15 b(a)984 5142 y Fh(3)1024 5128 y Fk(;)g(a)1112 5142 y Fh(4)1152 5128 y Fk(;)g(a)1240 5142 y Fh(5)1280 5128 y Fi(g)35 b Fp(of)g(mark)m(ed)g(ro)m(ws,)h(and)e(the)g(set)i Fk(U)2587 5142 y Fj(c)2654 5128 y Fp(=)c Fi(f)p Fk(a)2850 5142 y Fh(2)2890 5128 y Fi(g)j Fp(of)g(mark)m(ed)382 5241 y(columns.)k(Consequen)m(tly)-8 b(,)31 b(the)f(maxim)m(um)f(an)m (tipath)i Fi(U)39 b Fp(is)29 b(equal)h(to)1079 5403 y Fi(U)k Fp(=)25 b Fi(f)p Fk(a)1359 5417 y Fh(2)1399 5403 y Fk(;)15 b(a)1487 5417 y Fh(3)1527 5403 y Fk(;)g(a)1615 5417 y Fh(4)1655 5403 y Fk(;)g(a)1743 5417 y Fh(5)1783 5403 y Fi(g)21 b(n)f(f)p Fk(a)2007 5417 y Fh(2)2047 5403 y Fi(g)26 b Fp(=)f Fi(f)p Fk(a)2307 5417 y Fh(3)2347 5403 y Fk(;)15 b(a)2435 5417 y Fh(4)2475 5403 y Fk(;)g(a)2563 5417 y Fh(5)2603 5403 y Fi(g)p Fk(:)1854 5652 y Fp(6)p eop %%Page: 7 7 7 6 bop 523 548 a Fp(The)33 b(describ)s(ed)e(ab)s(o)m(v)m(e)k(F)-8 b(ord-F)g(ulk)m(erson)34 b(metho)s(dology)g(for)f(\014nding)e(an)m (tipaths)382 661 y(ma)m(y)h(b)s(e)f(easily)g(adapted)h(to)h(an)m(y)f (algorithm)f(\014nding)e(the)j(maxim)m(um)f(matc)m(hing)h(in)382 774 y(a)f(bipartite)f(graph,)h(for)g(example,)h(the)f(Hop)s(croft-Karp) g(algorithm)f([4)q(],)i([8],)g(or)f(the)382 887 y(\015o)m(w)j (algorithm)f([5)q(].)52 b(Therefore,)35 b(w)m(e)f(assume)g(that)h(the)f (running)d(time)j(of)g(a)h(MPP)382 1000 y(construction)30 b(is)f(equal)h(to)h Fk(O)s Fp(\()p Fk(n)1507 967 y Fh(5)p Fj(=)p Fh(2)1617 1000 y Fp(\).)523 1112 y(It)23 b(is)g(easy)g(to)h(see) g(that)g(the)f(Minim)m(um)e(P)m(ath)j(P)m(artition)f(Problem)f(for)h(a) h(partially)382 1225 y(ordered)d(set)g(and)g(the)g(Minim)m(um)e(Clique) g(P)m(artition)i(Problem)f(for)h(graphs)g(of)g Fk(L)3136 1239 y Fj(n)3204 1225 y Fp(lo)s(ok)382 1338 y(alik)m(e.)55 b(In)35 b(particular,)g(if)f(the)h(graph)g Fk(G)e Fi(2)g Fk(L)1987 1352 y Fj(n)2069 1338 y Fp(has)i(a)h(transitiv)m(e)f(orien)m (tation)g(then)382 1451 y(they)d(are)h(equiv)-5 b(alen)m(t.)46 b(Unfortunately)-8 b(,)32 b(in)f(general)i(case)g(it)f(is)f(false.)46 b(Ho)m(w)m(ev)m(er,)35 b(w)m(e)382 1564 y(can)c(use)f(this)f(similarit) m(y)f(b)s(elo)m(w)h(to)i(design)f(the)g(solving)f(algorithm.)382 1850 y Fl(3)135 b(The)44 b(Basic)h(De\014nitions)382 2053 y Fp(Recall)30 b(some)h(de\014nitions)d(of)i(Graph)g(Theory)g(and) f(in)m(tro)s(duce)h(new)g(ones.)523 2166 y(Let)37 b(there)f(b)s(e)f(a)i (graph)e Fk(G)f Fp(=)h(\()p Fk(X)r(;)15 b Fp(\000\))36 b Fi(2)e Fk(L)2053 2180 y Fj(n)2100 2166 y Fp(.)58 b(A)36 b(v)m(ertex)h(set)g Fk(U)44 b Fi(\022)35 b Fk(X)43 b Fp(is)35 b(called)382 2279 y Fg(indep)-5 b(endent)32 b Fp(if)1581 2392 y Fk(U)f Fi(\\)19 b Fp(\000\()p Fk(U)10 b Fp(\))26 b(=)f Fi(\013)p Fk(:)382 2557 y Fp(An)30 b(indep)s(enden)m (t)e(set)j Fk(U)40 b Fp(of)31 b(graph)f(v)m(ertices)h(is)e(called)h Fg(maximal)k(\(MIS\))c Fp(if)1578 2758 y Fk(U)g Fi([)20 b Fp(\000\()p Fk(U)10 b Fp(\))26 b(=)f Fk(X)r(:)523 2959 y Fp(A)38 b(MIS)850 2936 y(~)834 2959 y Fk(U)48 b Fp(is)38 b(called)f Fg(maximum)k(\(MMIS\))d Fp(if)f Fk(C)7 b(ar)s(d)p Fp(\()2452 2936 y(~)2436 2959 y Fk(U)j Fp(\))39 b Fi(\025)f Fk(C)7 b(ar)s(d)p Fp(\()p Fk(U)j Fp(\))38 b(for)g(an)m(y)382 3072 y(MIS)30 b Fk(U)40 b Fp(of)31 b Fk(G)p Fp(.)523 3185 y(P)m(artition)f(the)g(v)m(ertex)i(set)f Fk(X)38 b Fp(in)m(to)30 b(subsets)1559 3386 y Fk(X)1641 3348 y Fh(0)1681 3386 y Fk(;)15 b(X)1803 3348 y Fh(1)1843 3386 y Fk(;)g(:)g(:)g(:)h(;)f(X)2126 3348 y Fj(m)3255 3386 y Fp(\(1\))382 3587 y(in)36 b(suc)m(h)h(a)h(w)m(a)m(y)h(that)f(a)g (subset)f Fk(X)1637 3554 y Fj(k)1717 3587 y Fp(\()p Fk(i)h Fp(=)p 1929 3518 166 4 v 37 w(0)p Fk(;)15 b(m)q Fp(\))37 b(is)g(a)h(MIS)f(of)h(a)g(subgraph)d Fk(G)3220 3602 y Fj(k)3300 3587 y Fp(=)382 3700 y(\()p Fk(X)d Fi(n)24 b Fp(\()p Fk(X)710 3667 y Fh(0)775 3700 y Fi([)g Fk(X)942 3667 y Fh(1)1006 3700 y Fi([)f(\001)15 b(\001)g(\001)26 b([)d Fk(X)1387 3667 y Fj(k)r Fv(\000)p Fh(1)1521 3700 y Fp(\),)38 b(\000)1676 3715 y Fj(k)1719 3700 y Fp(\))e(=)g(\()p Fk(X)2014 3667 y Fj(k)2082 3700 y Fi([)24 b(\001)15 b(\001)g(\001)25 b([)e Fk(X)2463 3667 y Fj(m)2531 3700 y Fp(,)38 b(\000)2651 3715 y Fj(k)2693 3700 y Fp(\).)59 b(Clearly)-8 b(,)37 b Fk(G)3225 3714 y Fh(0)3300 3700 y Fp(=)382 3812 y(\()p Fk(X)499 3779 y Fh(0)559 3812 y Fi([)20 b(\001)15 b(\001)g(\001)21 b([)f Fk(X)929 3779 y Fj(m)996 3812 y Fp(,)31 b(\000)1109 3826 y Fh(0)1148 3812 y Fp(\))g(=)f Fk(G)p Fp(.)523 3925 y(By)e(the)h(giv)m(en)f(undirected)e(graph)i Fk(G)f Fp(and)h(the)g (partition)f(\(1\))i(w)m(e)g(can)f(construct)382 4038 y(a)40 b(digraph)824 4015 y Fk(~)810 4038 y(G)p Fp(\()p Fk(X)999 4005 y Fh(0)1039 4038 y Fp(\))g(=)f(\()p Fk(X)r(;)1375 4015 y(~)1376 4038 y Fp(\000)q(\))h(in)f(the)h(follo)m(wing)e(w)m(a)m (y)-8 b(.)70 b(If)39 b(an)h(edge)h(of)f Fk(G)f Fp(joins)f(a)382 4151 y(v)m(ertex)29 b Fk(x)707 4165 y Fj(i)761 4151 y Fi(2)24 b Fk(X)928 4118 y Fj(k)965 4127 y Fe(1)1033 4151 y Fp(with)i(a)j(v)m(ertex)g Fk(x)1636 4165 y Fj(j)1698 4151 y Fi(2)c Fk(X)1866 4118 y Fj(k)1903 4127 y Fe(2)1970 4151 y Fp(then)j(this)f(edge)i(is)e(replaced)h(b)m(y)g(an)g(arc)382 4264 y(\()p Fk(x)469 4278 y Fj(i)497 4264 y Fk(;)15 b(x)589 4278 y Fj(j)626 4264 y Fp(\))36 b(when)f Fk(k)987 4278 y Fh(1)1062 4264 y Fk(<)f(k)1214 4278 y Fh(2)1254 4264 y Fp(.)57 b(The)35 b(v)m(ertex)i Fk(x)1861 4278 y Fj(i)1925 4264 y Fp(is)e(called)g(the)h Fg(tail)h Fp(of)f(\()p Fk(x)2807 4278 y Fj(i)2835 4264 y Fk(;)15 b(x)2927 4278 y Fj(j)2964 4264 y Fp(\),)38 b(and)d(the)382 4377 y(v)m(ertex)c Fk(x)709 4391 y Fj(j)776 4377 y Fp(is)f(called)f(the)i Fg(he)-5 b(ad)32 b Fp(of)e(this)f(arc.)523 4500 y(As)34 b(the)f(result)g(w)m(e)h(ha)m(v)m(e)h(an)e(acyclic)h(digraph)2205 4477 y Fk(~)2191 4500 y(G)p Fp(\()p Fk(X)2380 4467 y Fh(0)2420 4500 y Fp(\))g(=)f(\()p Fk(X)r(;)2744 4477 y(~)2745 4500 y Fp(\000)q(\).)51 b(The)33 b(set)h Fk(X)3331 4467 y Fh(0)382 4613 y Fp(is)29 b(called)h Fg(initiating)p Fp(.)523 4726 y(In)d(general)g(case)h(w)m(e)g(can)g(construct)f(a)h (set)g Fk(D)s Fp(\()p Fk(G)p Fp(\))f(of)g(di\013eren)m(t)g(acyclic)h (digraphs)382 4839 y(eac)m(h)d(of)f(whic)m(h)f(is)g(generated)i(b)m(y)f (some)g(orien)m(tation)g(of)h(edges)f(of)g(the)h(graph)e Fk(G)i Fi(2)g Fk(L)3299 4853 y Fj(n)3345 4839 y Fp(.)382 4951 y(F)-8 b(urther)30 b(w)m(e)h(will)d(consider)h(only)g(digraphs)g (of)h Fk(D)s Fp(\()p Fk(G)p Fp(\).)523 5064 y(The)25 b(maxim)m(um)e(length)i Fk(\032)p Fp(\()p Fk(x)p Fp(\))g(of)h(a)f (directed)f(path,)i(connecting)g(a)f(v)m(ertex)h Fk(x)f Fi(2)g Fk(X)382 5177 y Fp(with)40 b(the)i(initiating)e(set)i Fk(X)1409 5144 y Fh(0)1449 5177 y Fp(,)i(is)d(called)g(the)h Fg(r)-5 b(ank)43 b Fp(of)f Fk(x)p Fp(.)74 b(The)41 b(set)i(of)f(all)e (graph)382 5290 y(v)m(ertices)e(ha)m(ving)g(the)f(same)i(rank)e Fk(\032)p Fp(\()p Fk(x)p Fp(\))h(=)f Fk(k)k Fp(is)c(called)g(the)g Fk(k)s Fg(-th)j(layer)f Fp(of)3120 5267 y Fk(~)3107 5290 y(G)o Fp(\()p Fk(X)3295 5257 y Fh(0)3335 5290 y Fp(\))382 5403 y(and)30 b(designated)g(as)g Fk(X)1202 5370 y Fj(k)1245 5403 y Fp(.)1854 5652 y(7)p eop %%Page: 8 8 8 7 bop 523 548 a Fp(A)26 b(directed)f(subgraph)1364 525 y Fk(~)1351 548 y(G)o Fp(\()p Fk(X)1539 515 y Fj(k)1583 548 y Fp(\))g(=)h(\()p Fk(X)1857 515 y Fj(k)1911 548 y Fi([)11 b Fk(X)2065 515 y Fj(k)r Fh(+1)2208 548 y Fi([)g(\001)k(\001) g(\001)c([)g Fk(X)2550 515 y Fj(m)2616 548 y Fk(;)2655 525 y(~)2656 548 y Fp(\000)2713 563 y Fj(k)2756 548 y Fp(\))26 b(of)g(the)f(digraph)395 638 y Fk(~)382 661 y(G)o Fp(\()p Fk(X)570 628 y Fh(0)611 661 y Fp(\))j(is)g(called)f(the)i (subgraph)d Fg(induc)-5 b(e)g(d)29 b Fp(b)m(y)g(the)f(la)m(y)m(er)h Fk(X)2468 628 y Fj(k)2511 661 y Fp(.)40 b(The)28 b(set)h Fk(X)2983 628 y Fj(k)3054 661 y Fp(is)e(called)382 774 y Fg(initiating)j Fp(for)g(the)g(directed)f(subgraph)1824 751 y Fk(~)1811 774 y(G)o Fp(\()p Fk(X)1999 741 y Fj(k)2043 774 y Fp(\).)41 b(A)30 b(digraph)2588 751 y Fk(~)2575 774 y(G)o Fp(\()p Fk(X)2763 741 y Fh(0)2803 774 y Fp(\))g(is)f(called)g Fg(c)-5 b(or-)382 887 y(r)g(e)g(ctly)35 b(c)-5 b(onstructe)g(d)35 b(\(CCD\))c Fp(if)g(in)f(eac)m(h)i(directed)f(subgraph)2564 864 y Fk(~)2551 887 y(G)o Fp(\()p Fk(X)2739 854 y Fj(k)2783 887 y Fp(\))g(the)h(initiating)382 1000 y(set)f Fk(X)606 967 y Fj(k)679 1000 y Fp(is)f(a)g(MIS)g(of)h(this)e(subgraph.)523 1112 y(T)-8 b(o)45 b(apply)e(the)h(partially)e(ordered)i(set)h(tec)m (hnique,)i(eac)m(h)f(digraph)3015 1090 y Fk(~)3001 1112 y(G)p Fp(\()p Fk(X)3190 1080 y Fh(0)3230 1112 y Fp(\))f(is)382 1225 y(assigned)27 b(to)i(a)f(transitiv)m(e)f(closure)h(graph)f (\(TCG\))2195 1202 y Fk(~)2181 1225 y(G)2253 1239 y Fj(t)2282 1225 y Fp(\()p Fk(X)2399 1192 y Fh(0)2440 1225 y Fp(\))h(=)f(\()p Fk(X)r(;)2752 1202 y(~)2753 1225 y Fp(\000)2810 1239 y Fj(t)2841 1225 y Fp(\))h([8)q(,)g(9)q(].)40 b(Since)382 1338 y(the)29 b(digraph)883 1315 y Fk(~)869 1338 y(G)p Fp(\()p Fk(X)1058 1305 y Fh(0)1098 1338 y Fp(\))g(is)f(acyclic)h(and)f (lo)s(opless,)g(its)g(transitiv)m(e)h(closure)2926 1315 y Fk(~)2913 1338 y(G)2985 1352 y Fj(t)3014 1338 y Fp(\()p Fk(X)3131 1305 y Fh(0)3171 1338 y Fp(\))g(is)f(a)382 1451 y(graph)33 b(of)h(a)g(strict)g(partial)e(order)i(\()p Fk(X)r(;)15 b(>)p Fp(\).)52 b(F)-8 b(urther,)34 b(w)m(e)g(will)d(not)j (distinguish)c(the)382 1564 y(transitiv)m(e)24 b(closure)g(graph)1345 1541 y Fk(~)1331 1564 y(G)1403 1578 y Fj(t)1432 1564 y Fp(\()p Fk(X)1549 1531 y Fh(0)1590 1564 y Fp(\))h(and)f(partially)e (ordered)i(set)i(\()p Fk(X)r(;)15 b(>)p Fp(\).)40 b(Therefore,)382 1677 y(w)m(e)31 b(will)c(consider,)j(for)g(example,)g(an)m(tipaths)g (of)h(the)g(TCG)2496 1654 y Fk(~)2483 1677 y(G)2555 1691 y Fj(t)2584 1677 y Fp(\()p Fk(X)2701 1644 y Fh(0)2741 1677 y Fp(\).)523 1790 y(There)39 b(exists)g(an)g(e\016cien)m(t)h (algorithm)e(to)i(construct)g(the)f(TCG.)g(Its)g(running)382 1903 y(time)30 b(is)f(equal)h(to)h Fk(O)s Fp(\()p Fk(n)1193 1870 y Fh(3)1233 1903 y Fp(\))f(\(see,)i(for)e(example,)g([8)q(,)h (7]\).)523 2016 y(An)i(arc)i(\()p Fk(x)918 2030 y Fj(i)946 2016 y Fk(;)15 b(x)1038 2030 y Fj(j)1075 2016 y Fp(\))34 b(of)1265 1993 y Fk(~)1251 2016 y(G)1323 2030 y Fj(t)1352 2016 y Fp(\()p Fk(X)1469 1983 y Fh(0)1509 2016 y Fp(\))g(will)d(b)s(e)j (called)f Fg(essential)h Fp(if)f(there)h(exists)f(the)h(arc)382 2139 y(\()p Fk(x)469 2153 y Fj(i)497 2139 y Fk(;)15 b(x)589 2153 y Fj(j)626 2139 y Fp(\))43 b(of)f(the)h(digraph)1347 2116 y Fk(~)1334 2139 y(G)o Fp(\()p Fk(X)1522 2106 y Fh(0)1562 2139 y Fp(\).)77 b(Otherwise,)45 b(the)d(arc)h(\()p Fk(x)2589 2153 y Fj(i)2617 2139 y Fk(;)15 b(x)2709 2153 y Fj(j)2746 2139 y Fp(\))43 b(will)d(b)s(e)h(called)382 2251 y Fg(\014ctitious)p Fp(.)g(An)29 b(essen)m(tial)g(arc)h(is)e(also) i(designated)f(as)h Fk(x)2333 2265 y Fj(i)2386 2251 y Fk(>)25 b(x)2534 2265 y Fj(j)2570 2251 y Fp(,)30 b(and)f(a)h (\014ctitious)e(arc)382 2364 y(is)j(also)h(designated)f(as)h Fk(x)1275 2378 y Fj(i)1331 2364 y Fi(\035)c Fk(x)1502 2378 y Fj(j)1539 2364 y Fp(.)45 b(A)32 b(\014ctitious)f(arc)h Fk(x)2290 2378 y Fj(i)2346 2364 y Fi(\035)c Fk(x)2517 2378 y Fj(j)2585 2364 y Fp(is)j(called)g Fg(elementary)382 2477 y Fp(if)38 b(there)h(is)f(a)h(path)g(of)g(length)f(2)i(from)e Fk(x)1878 2491 y Fj(i)1945 2477 y Fp(to)i Fk(x)2117 2491 y Fj(j)2153 2477 y Fp(.)67 b(Arcs)39 b Fk(x)2516 2491 y Fj(i)2583 2477 y Fk(>)h(x)2746 2492 y Fj(k)2827 2477 y Fp(and)f Fk(x)3065 2492 y Fj(k)3147 2477 y Fk(>)g(x)3309 2491 y Fj(j)3345 2477 y Fp(,)382 2590 y(comp)s(osing)29 b(this)h(path,)g(are)h(called)e Fg(gener)-5 b(atric)g(es)32 b Fp(for)e(the)h(\014ctitious)e(arc)i Fk(x)3027 2604 y Fj(i)3080 2590 y Fi(\035)25 b Fk(x)3248 2604 y Fj(j)3285 2590 y Fp(.)523 2713 y(Since)777 2690 y Fk(~)763 2713 y(G)835 2727 y Fj(t)864 2713 y Fp(\()p Fk(X)981 2680 y Fh(0)1022 2713 y Fp(\))33 b(is)g(a)g(graph)g(of)g(a)h(strict)f (partial)f(order,)i(w)m(e)g(can)f(\014nd)f(the)i(MPP)382 2826 y Fi(P)k Fp(=)30 b Fi(f)p Fk(S)685 2840 y Fh(1)725 2826 y Fk(;)15 b(:)g(:)g(:)h(;)f(S)982 2840 y Fj(p)1022 2826 y Fi(g)p Fp(.)41 b(In)30 b(general)g(case)i(this)d(partition)g(is) g(am)m(biguous.)523 2939 y(Di\013eren)m(t)34 b(MPPs)g(of)g(the)g (transitiv)m(e)f(closure)h(graph)f(are)h(sho)m(wn)g(in)e(Fig.)51 b(2)35 b(\(a\))382 3052 y(and)22 b(\(b\).)38 b(Arcs)23 b(of)f(the)h(digraph,)g(b)s(elonging)e(to)i(paths)f(of)h(the)f(MPP)-8 b(,)24 b(are)f(represen)m(ted)382 3165 y(b)m(y)30 b(thic)m(k)g(lines.) 523 3278 y(Let)39 b Fk(X)7 b Fp(\()p Fk(S)867 3292 y Fj(q)905 3278 y Fp(\))38 b(\()p Fk(q)k Fp(=)p 1204 3209 132 4 v 37 w(1)p Fk(;)15 b(p)q Fp(\))38 b(b)s(e)g(the)g(set)h(of)f(v)m (ertices,)j(b)s(elonging)36 b(to)j(the)f(path)g Fk(S)3308 3292 y Fj(q)3345 3278 y Fp(.)382 3390 y(If)k(v)m(ertices)h Fk(x)878 3404 y Fj(i)906 3390 y Fk(;)15 b(x)998 3404 y Fj(j)1080 3390 y Fi(2)44 b Fk(X)7 b Fp(\()p Fk(S)1358 3404 y Fj(q)1397 3390 y Fp(\))42 b(are)h(terminal)e(for)h(a)g (\014ctitious)g(arc)g Fk(x)2854 3404 y Fj(i)2927 3390 y Fi(\035)j Fk(x)3115 3404 y Fj(j)3194 3390 y Fp(then)382 3513 y(the)33 b(v)m(ertex)h Fk(x)871 3527 y Fj(j)940 3513 y Fp(is)e(called)g Fg(marke)-5 b(d)p Fp(.)49 b(The)33 b(set)g(of)g(all)f(mark)m(ed)h(v)m(ertices)g(of)2997 3490 y Fk(~)2983 3513 y(G)3055 3527 y Fj(t)3084 3513 y Fp(\()p Fk(X)3201 3480 y Fh(0)3241 3513 y Fp(\))h(is)382 3626 y(determined)28 b(b)m(y)i(the)g(found)f(MPP)h Fi(P)37 b Fp(and)29 b(it)h(di\013ers)e(for)i(di\013eren)m(t)f(MPPs.)41 b(The)29 b(set)382 3739 y(of)h(all)g(mark)m(ed)g(v)m(ertices)h(of)1381 3716 y Fk(~)1368 3739 y(G)1440 3753 y Fj(t)1469 3739 y Fp(\()p Fk(X)1586 3706 y Fh(0)1626 3739 y Fp(\))g(is)e(designated)h (as)h Fi(B)s Fp(\()p Fi(P)7 b Fp(\).)523 3852 y(F)-8 b(or)33 b(example,)f(for)g(the)g(MPP)g Fi(P)1672 3866 y Fh(1)1744 3852 y Fp(represen)m(ted)g(in)e(Fig.)46 b(2)32 b(\(a\),)i(w)m(e)f(ha)m(v)m(e)g Fi(B)s Fp(\()p Fi(P)3296 3866 y Fh(1)3335 3852 y Fp(\))382 3965 y(=)g Fi(f)p Fk(x)583 3979 y Fh(5)623 3965 y Fk(;)15 b(x)715 3979 y Fh(6)755 3965 y Fi(g)p Fp(,)36 b(and)d(for)h(the)g(MPP)g Fi(P)1648 3979 y Fh(2)1721 3965 y Fp(represen)m(ted)g(in)f(Fig.)51 b(2)34 b(\(b\),)h(w)m(e)g(ha)m(v)m(e)g Fi(B)s Fp(\()p Fi(P)3296 3979 y Fh(2)3335 3965 y Fp(\))382 4078 y(=)30 b Fi(\013)p Fp(.)382 4290 y Fq(Lemma)j(1)45 b Fg(L)-5 b(et)1032 4267 y Fk(~)1018 4290 y(G)p Fp(\()p Fk(X)1207 4257 y Fh(0)1247 4290 y Fp(\))43 b Fg(b)-5 b(e)44 b(a)g(digr)-5 b(aph,)48 b(and)2102 4267 y Fk(~)2088 4290 y(G)2160 4304 y Fj(t)2189 4290 y Fp(\()p Fk(X)2306 4257 y Fh(0)2347 4290 y Fp(\))43 b Fg(b)-5 b(e)44 b(its)f(tr)-5 b(ansitive)45 b(closur)-5 b(e)382 4403 y(gr)g(aph.)64 b(If)40 b Fi(B)s Fp(\()p Fi(P)7 b Fp(\))38 b(=)g Fi(\013)p Fg(,)i(wher)-5 b(e)41 b Fi(P)47 b Fg(is)40 b(an)g(MPP)e(of)2266 4380 y Fk(~)2253 4403 y(G)2325 4417 y Fj(t)2354 4403 y Fp(\()p Fk(X)2471 4370 y Fh(0)2511 4403 y Fp(\))p Fg(,)j(then)f(the)g(maximum) 382 4516 y(antip)-5 b(ath)35 b Fi(U)42 b Fg(is)33 b(the)g(MMIS)f(of)h (the)g(gr)-5 b(aph)34 b Fk(G)25 b Fi(2)h Fk(L)2116 4530 y Fj(n)2162 4516 y Fg(,)33 b(and)h(the)f(MPP)f(determines)h(the)382 4629 y(minimum)g(clique)f(p)-5 b(artition)35 b(of)e Fk(G)p Fg(.)523 4842 y Fp(If)c(conditions)f(of)i(Lemma)g(1)g(are)g (satis\014ed)f(then)g(eac)m(h)i(path)e Fk(S)2726 4856 y Fj(q)2789 4842 y Fi(2)c(P)37 b Fp(is)29 b(a)h(clique)382 4964 y(of)499 4941 y Fk(~)485 4964 y(G)p Fp(\()p Fk(X)674 4931 y Fh(0)714 4964 y Fp(\).)41 b(Therefore,)31 b(the)f(MPP)g Fi(P)38 b Fp(is)30 b(the)g(minim)m(um)e(clique)h(partition.)523 5077 y(On)20 b(the)h(other)g(hand,)h(v)m(ertices)f(of)g(the)g(maxim)m (um)e(an)m(tipath)i Fi(U)29 b Fp(of)2810 5054 y Fk(~)2796 5077 y(G)2868 5091 y Fj(t)2897 5077 y Fp(\()p Fk(X)3014 5044 y Fh(0)3055 5077 y Fp(\))21 b(b)s(elong)382 5190 y(to)34 b(distinct)d(cliques,)i(that)g(is,)g(the)h(n)m(um)m(b)s(er)d (of)j(v)m(ertices)f(in)f(the)h(MMIS)g(is)f(equal)h(to)382 5303 y(the)d(n)m(um)m(b)s(er)g(of)g(v)m(ertices)h(in)e(the)i(set)g Fi(U)9 b Fp(.)1295 b(Q.E.D.)1854 5652 y(8)p eop %%Page: 9 9 9 8 bop 980 2650 a 3.3 setlinewidth 1158 2414 4 591 v 1159 1825 591 4 v 1748 2414 4 591 v 1159 2416 591 4 v 1159 1943 V 1159 2060 V 1159 2179 V 1159 2296 V 1276 2414 4 591 v 1394 2414 V 1512 2414 V 1630 2414 V 1314 1910 a Fp(1)1336 1880 y Fd(j)1432 1910 y Fp(1)191 b(1)1314 2264 y(1)1314 2382 y(1)1045 1895 y Fk(a)1093 1909 y Fh(1)1045 2013 y Fk(a)1093 2027 y Fh(2)1045 2131 y Fk(a)1093 2145 y Fh(3)1045 2249 y Fk(a)1093 2263 y Fh(4)1045 2368 y Fk(a)1093 2382 y Fh(5)1163 1765 y Fk(a)1211 1779 y Fh(1)1281 1765 y Fk(a)1329 1779 y Fh(2)1401 1765 y Fk(a)1449 1779 y Fh(3)1517 1765 y Fk(a)1565 1779 y Fh(4)1637 1765 y Fk(a)1685 1779 y Fh(5)p 2102 2414 V 2104 1825 591 4 v 2693 2414 4 591 v 2104 2416 591 4 v 2104 1943 V 2104 2061 V 2104 2179 V 2104 2297 V 2220 2414 4 591 v 2339 2414 V 2457 2414 V 2575 2414 V 2258 1910 a Fp(1)74 b(1)2399 1880 y Fd(j)2613 1910 y Fp(1)2258 2264 y(1)2281 2234 y Fd(j)2258 2382 y Fp(1)1989 1895 y Fk(a)2037 1909 y Fh(1)1989 2013 y Fk(a)2037 2027 y Fh(2)1989 2131 y Fk(a)2037 2145 y Fh(3)1989 2249 y Fk(a)2037 2263 y Fh(4)1989 2368 y Fk(a)2037 2382 y Fh(5)2119 1765 y Fk(a)2167 1779 y Fh(1)2237 1765 y Fk(a)2285 1779 y Fh(2)2356 1765 y Fk(a)2404 1779 y Fh(3)2474 1765 y Fk(a)2522 1779 y Fh(4)2592 1765 y Fk(a)2640 1779 y Fh(5)1798 2022 y Fi(\003)1798 2140 y(\003)1798 2258 y(\003)1798 2376 y(\003)1432 2514 y Fp(1)191 b(1)-399 b(4)1798 1912 y(2)2719 2022 y Fi(\003)2719 2140 y(\003)2719 2266 y Fp(2)2719 2376 y Fi(\003)2258 2514 y Fp(5)1382 2649 y(\(b\))805 b(\(c\))p 1819 1469 4 473 v 1765 1080 a Fc(\012)1710 1163 y(\012)1663 1233 y(\012)1663 1316 y(J)1718 1399 y(J)1765 1469 y(J)q(\001)1862 1386 y(\001)1904 1303 y(\001)1945 1220 y(\001)1987 1137 y(\001)2015 1080 y(\001)1824 1473 y(\001)1866 1390 y(\001)1907 1307 y(\001)1949 1223 y(\001)1990 1140 y(\001)2019 1083 y(\001)1828 1476 y(\001)1869 1393 y(\001)1911 1310 y(\001)1952 1227 y(\001)1994 1144 y(\001)2022 1087 y(\001)1817 1465 y(\001)1859 1382 y(\001)1900 1299 y(\001)1942 1216 y(\001)1983 1133 y(\001)2012 1076 y(\001)1813 1462 y(\001)1855 1379 y(\001)1897 1296 y(\001)1938 1213 y(\001)1980 1130 y(\001)2008 1073 y(\001)1821 1080 y(A)1862 1163 y(A)1904 1246 y(A)1945 1329 y(A)1987 1412 y(A)2015 1469 y(A)1824 1076 y(A)1866 1159 y(A)1907 1242 y(A)1949 1325 y(A)1990 1408 y(A)2019 1465 y(A)1828 1073 y(A)1869 1156 y(A)1911 1239 y(A)1952 1322 y(A)1994 1405 y(A)2022 1462 y(A)1817 1083 y(A)1859 1166 y(A)1900 1249 y(A)1942 1332 y(A)1983 1415 y(A)2012 1473 y(A)1813 1087 y(A)1855 1170 y(A)1897 1253 y(A)1938 1336 y(A)1980 1419 y(A)2008 1476 y(A)1821 997 y Fd(x)161 b(x)1655 1233 y(x)1821 1469 y(x)g(x)1706 1553 y Fk(a)1754 1567 y Fh(1)1517 1269 y Fk(a)1565 1283 y Fh(5)1706 915 y Fk(a)1754 929 y Fh(2)2084 915 y Fk(a)2132 929 y Fh(3)2084 1553 y Fk(a)2132 1567 y Fh(4)1857 1657 y Fp(\(a\))p 1360 1922 71 4 v 1348 1920 a Fc(-)p 1358 2227 4 308 v 1360 2227 a(?)1414 2846 y Fp(Fig.)31 b(1:)41 b(The)30 b(ordered)g(set)1169 4744 y 3.3 setlinewidth 1346 4506 4 591 v 1348 3999 a Fc(J)1403 4082 y(J)1459 4165 y(J)1514 4248 y(J)1570 4331 y(J)1625 4414 y(J)1680 4497 y(J)1694 4518 y(J)1352 3995 y(J)1407 4078 y(J)1462 4161 y(J)1518 4244 y(J)1573 4327 y(J)1628 4410 y(J)1684 4493 y(J)1698 4514 y(J)1355 3992 y(J)1411 4075 y(J)1466 4158 y(J)1521 4241 y(J)1577 4324 y(J)1632 4407 y(J)1687 4490 y(J)1701 4511 y(J)1345 4002 y(J)1400 4085 y(J)1455 4168 y(J)1511 4251 y(J)1566 4334 y(J)1621 4417 y(J)1677 4500 y(J)1691 4521 y(J)1341 4006 y(J)1396 4089 y(J)1452 4172 y(J)1507 4255 y(J)1562 4338 y(J)1618 4421 y(J)1673 4504 y(J)1687 4525 y(J)p 1742 4506 V 1688 3999 a(\012)1633 4082 y(\012)1577 4165 y(\012)1522 4248 y(\012)1467 4331 y(\012)1411 4414 y(\012)1356 4497 y(\012)1342 4518 y(\012)1692 4002 y(\012)1636 4085 y(\012)1581 4168 y(\012)1525 4251 y(\012)1470 4334 y(\012)1415 4417 y(\012)1359 4500 y(\012)1345 4521 y(\012)1695 4006 y(\012)1640 4089 y(\012)1584 4172 y(\012)1529 4255 y(\012)1474 4338 y(\012)1418 4421 y(\012)1363 4504 y(\012)1349 4525 y(\012)1687 3995 y(\012)1631 4078 y(\012)1576 4161 y(\012)1521 4244 y(\012)1465 4327 y(\012)1410 4410 y(\012)1355 4493 y(\012)1341 4514 y(\012)1683 3992 y(\012)1628 4075 y(\012)1573 4158 y(\012)1517 4241 y(\012)1462 4324 y(\012)1407 4407 y(\012)1351 4490 y(\012)1337 4511 y(\012)1348 4506 y(\003)1369 4423 y(\003)1390 4340 y(\003)1410 4257 y(\003)1431 4174 y(\003)1434 4161 y(\003)1729 4506 y(C)1708 4423 y(C)1687 4340 y(C)1667 4257 y(C)1646 4174 y(C)1643 4161 y(C)1348 3916 y Fd(x)327 b(x)1466 4081 y(x)91 b(x)1348 4506 y(x)327 b(x)1397 4590 y Fk(x)1449 4604 y Fh(1)1799 4590 y Fk(x)1851 4604 y Fh(2)1432 3999 y Fk(x)1484 4013 y Fh(3)1574 3999 y Fk(x)1626 4013 y Fh(4)1326 3834 y Fk(x)1378 3848 y Fh(5)1680 3834 y Fk(x)1732 3848 y Fh(6)p 2055 4506 V 2057 3999 a Fc(J)2112 4082 y(J)2167 4165 y(J)2223 4248 y(J)2278 4331 y(J)2334 4414 y(J)2389 4497 y(J)2403 4518 y(J)2060 3995 y(J)2116 4078 y(J)2123 4089 y(J)2064 3992 y(J)2119 4075 y(J)2127 4086 y(J)2053 4002 y(J)2109 4085 y(J)2116 4096 y(J)2050 4006 y(J)2105 4089 y(J)2112 4100 y(J)p 2450 4506 V 2397 3999 a(\012)2341 4082 y(\012)2286 4165 y(\012)2231 4248 y(\012)2175 4331 y(\012)2120 4414 y(\012)2065 4497 y(\012)2050 4518 y(\012)2400 4002 y(\012)2345 4085 y(\012)2337 4096 y(\012)2404 4006 y(\012)2348 4089 y(\012)2341 4100 y(\012)2393 3995 y(\012)2338 4078 y(\012)2330 4089 y(\012)2390 3992 y(\012)2334 4075 y(\012)2327 4086 y(\012)2057 4506 y(\003)2078 4423 y(\003)2098 4340 y(\003)2119 4257 y(\003)2140 4174 y(\003)2143 4161 y(\003)2060 4510 y(\003)2081 4427 y(\003)2102 4344 y(\003)2123 4261 y(\003)2143 4178 y(\003)2147 4165 y(\003)2064 4513 y(\003)2085 4430 y(\003)2105 4347 y(\003)2126 4264 y(\003)2147 4181 y(\003)2150 4168 y(\003)2053 4503 y(\003)2074 4420 y(\003)2095 4337 y(\003)2116 4254 y(\003)2136 4170 y(\003)2140 4158 y(\003)2050 4499 y(\003)2070 4416 y(\003)2091 4333 y(\003)2112 4250 y(\003)2133 4167 y(\003)2136 4154 y(\003)2438 4506 y(C)2417 4423 y(C)2396 4340 y(C)2375 4257 y(C)2355 4174 y(C)2351 4161 y(C)2441 4503 y(C)2420 4420 y(C)2400 4337 y(C)2379 4254 y(C)2358 4170 y(C)2355 4158 y(C)2445 4499 y(C)2424 4416 y(C)2403 4333 y(C)2382 4250 y(C)2362 4167 y(C)2358 4154 y(C)2434 4510 y(C)2413 4427 y(C)2393 4344 y(C)2372 4261 y(C)2351 4178 y(C)2348 4165 y(C)2431 4513 y(C)2410 4430 y(C)2389 4347 y(C)2368 4264 y(C)2348 4181 y(C)2344 4168 y(C)2057 3916 y Fd(x)f(x)2175 4081 y(x)90 b(x)2057 4506 y(x)326 b(x)2106 4590 y Fk(x)2158 4604 y Fh(1)2507 4590 y Fk(x)2559 4604 y Fh(2)2141 3999 y Fk(x)2193 4013 y Fh(3)2283 3999 y Fk(x)2335 4013 y Fh(4)2035 3834 y Fk(x)2087 3848 y Fh(5)2389 3834 y Fk(x)2441 3848 y Fh(6)1481 4719 y Fp(\(a\))591 b(\(b\))804 4940 y(Fig.)31 b(2:)41 b(Di\013eren)m(t)31 b(MPPs)f(of)h(the)f(transitiv)m (e)g(closure)g(graph)1854 5652 y(9)p eop %%Page: 10 10 10 9 bop 523 548 a Fp(Notice)45 b(that)h(Lemma)e(1)h(ma)m(y)g(b)s(e)f (satis\014ed)f(if)h(the)g(digraph)2804 525 y Fk(~)2790 548 y(G)p Fp(\()p Fk(X)2979 515 y Fh(0)3019 548 y Fp(\))h(has)f(no)382 661 y(transitiv)m(e)30 b(orien)m(tation.)523 873 y(Let)43 b(us)f(clarify)f(the)i(nature)f(of)h(emerging)f(of)h(mark)m(ed)g(v)m (ertices,)j(taking)d(in)m(to)382 986 y(accoun)m(t)32 b(the)e(structure)g(of)h(a)g(graph)e Fk(G)c Fi(2)g Fk(L)1946 1000 y Fj(n)1993 986 y Fp(.)523 1099 y(In)37 b(a)h(digraph)d Fk(G)p Fp(\()p Fk(X)1256 1066 y Fh(0)1296 1099 y Fp(\))i Fi(2)g Fk(D)s Fp(\()p Fk(G)p Fp(\),)i(it)e(is)g(exp)s(edien)m(t)f(to)j (distinguish)33 b(undirected)382 1212 y(paths)24 b(and)g(cycles)g(if)f (they)i(corresp)s(ond)e(to)i(paths)f(and)g(cycles)g(of)h(the)f(graph)g Fk(G)h Fi(2)g Fk(L)3299 1226 y Fj(n)3345 1212 y Fp(.)523 1325 y(W)-8 b(e)26 b(will)c(sa)m(y)j(that)h(a)f(simple)d(undirected)h (cycle)j(or)e(a)h(simple)e(undirected)g(path)h(of)395 1415 y Fk(~)382 1438 y(G)o Fp(\()p Fk(X)570 1405 y Fh(0)611 1438 y Fp(\))j Fi(2)f Fk(D)s Fp(\()p Fk(G)p Fp(\))31 b(is)f Fg(str)-5 b(etche)g(d)34 b Fp(on)d(paths)g Fk(S)1912 1452 y Fh(1)1951 1438 y Fk(;)15 b(:)g(:)g(:)h(;)f(S)2208 1452 y Fj(p)2279 1438 y Fp(of)32 b(the)f(MPP)g Fi(P)39 b Fp(of)32 b(the)f(graph)395 1538 y Fk(~)382 1561 y(G)454 1575 y Fj(t)483 1561 y Fp(\()p Fk(X)600 1528 y Fh(0)640 1561 y Fp(\))j(if)e(eac)m(h)i(of)g(paths)f Fk(S)1416 1575 y Fj(r)1487 1561 y Fp(\()p Fk(r)f Fp(=)p 1696 1492 132 4 v 30 w(1)p Fk(;)15 b(p)q Fp(\))33 b(passes)g(through)g(at)h (least)g(one)f(v)m(ertex)h(of)382 1674 y(the)27 b(simple)d(undirected)h (cycle)i(\(path\))g(and)f(there)h(is)e(no)i(other)g(path)f(of)h(the)f (MPP)h Fi(P)382 1787 y Fp(passing)i(through)h(v)m(ertices)h(of)f(the)h (considered)e(simple)f(undirected)h(cycle)i(\(path\).)523 1899 y(Also,)i(w)m(e)g(will)d(sa)m(y)j(that)g(a)g(simple)e(undirected)f (cycle)j(\(path\))g(is)f Fg(densely)j(str)-5 b(et-)382 2012 y(che)g(d)37 b Fp(on)e(paths)h Fk(S)1031 2026 y Fj(r)1104 2012 y Fp(\()p Fk(r)h Fp(=)p 1322 1944 V 34 w(1)p Fk(;)15 b(p)q Fp(\))36 b(of)g(the)g(MPP)g Fi(P)43 b Fp(of)36 b(the)g(graph)2693 1989 y Fk(~)2679 2012 y(G)2751 2026 y Fj(t)2780 2012 y Fp(\()p Fk(X)2897 1979 y Fh(0)2938 2012 y Fp(\))g(if)f(no)g(t)m(w)m(o)382 2125 y(indep)s(enden)m(t)28 b(v)m(ertices)j(of)g(the)f(cycle)h(\(path\))g(b)s(elong)e(to)i(the)g (same)g(path)f Fk(S)3048 2139 y Fj(r)3085 2125 y Fp(.)523 2238 y(It)39 b(is)f(easy)h(to)g(see)h(that)f(a)g(simple)e(undirected)g (path)h(of)h(the)g(length)f Fk(l)i Fp(can)f(b)s(e)382 2351 y(densely)27 b(stretc)m(hed)i(on)f(not)g(less)f(than)h Fi(d)p Fk(l)r(=)p Fp(2)p Fi(e)18 b Fp(+)d(1)28 b(paths)g(of)g Fi(P)7 b Fp(,)30 b(a)e(simple)e(undirected)382 2464 y(ev)m(en)40 b(cycle)f Fk(C)896 2479 y Fh(2)p Fj(k)1013 2464 y Fp(\()p Fk(k)44 b Fi(\025)c Fp(2\))g(can)f(b)s(e)g(densely)f(stretc)m(hed)i(on) f Fk(k)j Fp(paths)d(of)g Fi(P)7 b Fp(,)43 b(and)38 b(a)382 2577 y(simple)24 b(undirected)g(o)s(dd)h(cycle)h Fk(C)1566 2592 y Fh(2)p Fj(k)r Fh(+1)1760 2577 y Fp(\()p Fk(k)j Fi(\025)c Fp(2\))i(can)f(b)s(e)f(densely)g(stretc)m(hed)i(on)f Fk(k)14 b Fp(+)d(1)382 2690 y(paths)30 b(of)g Fi(P)7 b Fp(.)523 2803 y(Ob)m(viously)-8 b(,)24 b(mark)m(ed)h(v)m(ertices)g (ma)m(y)g(exist)f(in)f(a)i(graph)2439 2780 y Fk(~)2426 2803 y(G)2498 2817 y Fj(t)2527 2803 y Fp(\()p Fk(X)2644 2770 y Fh(0)2684 2803 y Fp(\))g(only)e(when)g(there)382 2916 y(is)e(suc)m(h)g(an)h(undirected)e(cycle)i(or)g(a)g(path)g(that)g (is)f(stretc)m(hed)i(on)e(paths)h(of)g(the)g(existing)382 3029 y(MPP)30 b Fi(P)38 b Fp(of)31 b(this)e(graph)h(non-densely)-8 b(.)523 3141 y(W)g(e)34 b(notice)e(that)h(an)m(y)g(an)m(tipath)f Fk(U)39 b Fi(\032)29 b Fk(X)39 b Fp(of)2127 3118 y Fk(~)2113 3141 y(G)2185 3155 y Fj(t)2214 3141 y Fp(\()p Fk(X)2331 3108 y Fh(0)2371 3141 y Fp(\))33 b(determines)e(an)i(indep)s(en-)382 3254 y(den)m(t)26 b(v)m(ertex)g(set)g(of)g(the)g(digraph)1582 3231 y Fk(~)1568 3254 y(G)p Fp(\()p Fk(X)1757 3221 y Fh(0)1797 3254 y Fp(\),)h(and)e(an)g(indep)s(enden)m(t)e(v)m(ertex)k (set)f Fk(U)35 b Fi(\032)25 b Fk(X)382 3367 y Fp(of)497 3344 y Fk(~)483 3367 y(G)p Fp(\()p Fk(X)672 3334 y Fh(0)712 3367 y Fp(\))j(determines)f(an)h(an)m(tipath)g(of)1836 3344 y Fk(~)1822 3367 y(G)1894 3381 y Fj(t)1923 3367 y Fp(\()p Fk(X)2040 3334 y Fh(0)2080 3367 y Fp(\))h(if)e(and)g(only)g (if)g(no)h(t)m(w)m(o)i(v)m(ertices)e(of)382 3480 y Fk(U)40 b Fp(b)s(elong)30 b(to)h(the)f(same)h(directed)f(path)g(of)1948 3457 y Fk(~)1934 3480 y(G)p Fp(\()p Fk(X)2123 3447 y Fh(0)2163 3480 y Fp(\).)382 3767 y Fl(4)135 b(A)44 b(VS-digraph)382 3970 y Fp(Let)31 b(there)f(b)s(e)g(an)g(acyclic)h(digraph)1670 3947 y Fk(~)1656 3970 y(G)p Fp(\()p Fk(X)1845 3937 y Fh(0)1885 3970 y Fp(\))g(=)f(\()p Fk(X)r(;)2203 3947 y(~)2204 3970 y Fp(\000)q(\).)523 4082 y(F)-8 b(urther,)37 b(let)f Fk(W)47 b Fi(\032)34 b Fk(X)44 b Fp(b)s(e)35 b(some)h(indep)s(enden)m(t)e(v)m(ertex)j(set.)58 b(F)-8 b(or)36 b(the)g(digraph)395 4172 y Fk(~)382 4195 y(G)o Fp(\()p Fk(X)570 4162 y Fh(0)611 4195 y Fp(\))43 b(w)m(e)g(de\014ne)f (a)h(unary)f(op)s(eration)g Fg(cutting)g Fk(\033)2256 4209 y Fj(W)2337 4195 y Fp(\()2386 4172 y Fk(~)2372 4195 y(G)p Fp(\()p Fk(X)2561 4162 y Fh(0)2601 4195 y Fp(\)\).)78 b(This)41 b(op)s(eration)382 4308 y(consists)35 b(of)h(reorien)m (tation)g(of)g(all)e(arcs)i(of)1931 4285 y Fk(~)1917 4308 y(G)p Fp(\()p Fk(X)2106 4275 y Fh(0)2146 4308 y Fp(\))g(incoming)e(in)m(to)i(v)m(ertices)h(of)e(the)382 4421 y(set)h Fk(W)13 b Fp(.)54 b(It)36 b(is)e(easy)i(to)g(see)g(that)g (the)f(result)f(of)i(this)e(op)s(eration)g(is)h(also)g(a)g(digraph)395 4511 y Fk(~)382 4534 y(G)o Fp(\()p Fk(Y)562 4501 y Fh(0)601 4534 y Fp(\),)c(where)1355 4647 y Fk(Y)1428 4609 y Fh(0)1493 4647 y Fp(=)25 b(\()p Fk(X)1706 4609 y Fh(0)1766 4647 y Fi(n)1831 4624 y Fk(~)1832 4647 y Fp(\000)1889 4609 y Fv(\000)p Fh(1)1983 4647 y Fp(\()p Fk(W)13 b Fp(\)\))21 b Fi([)f Fk(W)m(:)382 4814 y Fp(Here,)621 4791 y Fk(~)622 4814 y Fp(\000)679 4781 y Fv(\000)p Fh(1)803 4814 y Fp(is)30 b(a)h(map,)f(in)m(v)m(erse)g(to)1609 4791 y Fk(~)1610 4814 y Fp(\000.)382 5026 y Fq(Theorem)k(2)46 b Fg(L)-5 b(et)38 b(ther)-5 b(e)39 b(b)-5 b(e)37 b(a)i(digr)-5 b(aph)1859 5003 y Fk(~)1846 5026 y(G)o Fp(\()p Fk(X)2034 4993 y Fh(0)2075 5026 y Fp(\))35 b Fi(2)f Fk(D)s Fp(\()p Fk(G)p Fp(\))k Fg(and)h Fk(W)50 b Fg(b)-5 b(e)38 b(some)h(inde-)382 5139 y(p)-5 b(endent)29 b(vertex)g(set.)40 b(Then)29 b(a)f(digr)-5 b(aph)1794 5116 y Fk(~)1781 5139 y(G)o Fp(\()p Fk(Y)1961 5106 y Fh(0)2000 5139 y Fp(\))28 b Fg(=)g Fk(\033)2213 5153 y Fj(W)2294 5139 y Fp(\()2343 5116 y Fk(~)2329 5139 y(G)p Fp(\()p Fk(X)2518 5106 y Fh(0)2558 5139 y Fp(\)\))h Fg(is)f(also)h(acyclic)g(and)395 5229 y Fk(~)382 5252 y(G)o Fp(\()p Fk(Y)562 5219 y Fh(0)601 5252 y Fp(\))d Fi(2)f Fk(D)s Fp(\()p Fk(G)p Fp(\))p Fg(.)1831 5652 y Fp(10)p eop %%Page: 11 11 11 10 bop 523 548 a Fp(Indeed,)38 b(since)f(the)g(digraph)1599 525 y Fk(~)1586 548 y(G)o Fp(\()p Fk(X)1774 515 y Fh(0)1814 548 y Fp(\))h(=)f(\()p Fk(X)r(;)2146 525 y(~)2147 548 y Fp(\000)q(\))g(is)f(acyclic)i(then)f(an)m(y)g(its)g(part)382 661 y(is)c(also)h(an)g(acyclic)g(digraph.)50 b(Th)m(us,)34 b(a)g(directed)g(subgraph)2576 638 y Fk(~)2562 661 y(G)2634 675 y Fh(1)2707 661 y Fp(=)g(\()p Fk(X)c Fi(n)23 b Fk(W)m(;)3142 638 y(~)3143 661 y Fp(\000)3200 675 y Fh(1)3240 661 y Fp(\))35 b(is)382 774 y(acyclic.)523 887 y(Ob)m(viously)-8 b(,)990 864 y Fk(~)976 887 y(G)1048 901 y Fh(1)1123 887 y Fi(\032)1243 864 y Fk(~)1230 887 y(G)o Fp(\()p Fk(Y)1410 854 y Fh(0)1449 887 y Fp(\).)61 b(A)m(ttac)m(h)39 b(the)e(indep)s (enden)m(t)d(v)m(ertex)k(set)g Fk(W)49 b Fp(to)37 b(the)382 1000 y(subgraph)785 977 y Fk(~)771 1000 y(G)843 1014 y Fh(1)882 1000 y Fp(.)i(Join)23 b(eac)m(h)j(v)m(ertex)g Fk(x)g Fi(2)f Fk(W)37 b Fp(with)23 b(a)j(v)m(ertex)g Fk(y)h Fp(of)2620 977 y Fk(~)2606 1000 y(G)2678 1014 y Fh(1)2742 1000 y Fp(b)m(y)e(the)g(arc)g(\()p Fk(x;)15 b(y)s Fp(\))382 1112 y(if)30 b(and)g(only)h(if)f(there)h(exists)f(the)i (arc)f(\()p Fk(y)s(;)15 b(x)p Fp(\))32 b(of)f(the)h(digraph)2573 1090 y Fk(~)2560 1112 y(G)o Fp(\()p Fk(X)2748 1080 y Fh(0)2789 1112 y Fp(\).)43 b(It)31 b(is)f(eviden)m(t)382 1225 y(that)h(the)f(resulting)f(digraph)1457 1202 y Fk(~)1444 1225 y(G)o Fp(\()p Fk(Y)1623 1192 y Fh(0)1663 1225 y Fp(\))h(is)g(also)g(acyclic.)523 1338 y(A)m(t)39 b(last,)g(w)m(e)f(ha)m (v)m(e)1240 1315 y Fk(~)1226 1338 y(G)p Fp(\()p Fk(Y)1406 1305 y Fh(0)1445 1338 y Fp(\))g Fi(2)f Fk(D)s Fp(\()p Fk(G)p Fp(\))g(since)g(an)m(y)h(reorien)m(tation)g(of)g(arcs)g(of)f (the)382 1451 y(digraph)725 1428 y Fk(~)712 1451 y(G)o Fp(\()p Fk(X)900 1418 y Fh(0)940 1451 y Fp(\))27 b(do)s(es)f(not)h(c)m (hange)h(indep)s(endence)c(relation)h(of)i(its)f(v)m(ertices.Q.E.D.)523 1664 y(Let)31 b(there)f(b)s(e)g(CCD)1287 1641 y Fk(~)1273 1664 y(G)p Fp(\()p Fk(X)1462 1631 y Fh(0)1502 1664 y Fp(\))h(and)e(its)h(transitiv)m(e)g(closure)g(graph)2851 1641 y Fk(~)2838 1664 y(G)2910 1678 y Fj(t)2939 1664 y Fp(\()p Fk(X)3056 1631 y Fh(0)3096 1664 y Fp(\).)523 1777 y(W)-8 b(e)26 b(can)f(\014nd)f(the)h(MPP)g Fi(P)32 b Fp(of)25 b(the)g(graph)2005 1754 y Fk(~)1992 1777 y(G)2064 1791 y Fj(t)2093 1777 y Fp(\()p Fk(X)2210 1744 y Fh(0)2250 1777 y Fp(\))g(constructing)g(the)g(maxim)m(um)382 1890 y(an)m(tipath)30 b Fi(U)39 b Fp(sim)m(ultaneously)28 b(as)j(it)f(w)m(as)h(describ)s(ed)d(in)h(Section)h(2.)523 2002 y(In)g(general)g(case)h(w)m(e)g(can)g(\014nd)e(some)i(distinct)d (maxim)m(um)i(an)m(tipaths.)523 2115 y(W)-8 b(e)38 b(will)c(sa)m(y)j (that)g(an)f(an)m(tipath)g Fk(U)1796 2129 y Fh(1)1872 2115 y Fg(pr)-5 b(e)g(c)g(e)g(des)39 b Fp(an)d(an)m(tipath)g Fk(U)2792 2129 y Fh(2)2868 2115 y Fp(in)g(the)g(graph)395 2205 y Fk(~)382 2228 y(G)454 2242 y Fj(t)483 2228 y Fp(\()p Fk(X)600 2195 y Fh(0)640 2228 y Fp(\))31 b(and)f(designate)g(it)g(as)h Fk(U)1547 2242 y Fh(1)1612 2228 y Fi(\036)25 b Fk(U)1770 2242 y Fh(2)1840 2228 y Fp(if)30 b(for)g(all)f(v)m(ertices)i Fk(x)26 b Fi(2)f Fk(U)2744 2242 y Fh(1)2803 2228 y Fi(n)c Fk(U)2931 2242 y Fh(2)3001 2228 y Fp(there)31 b(is)e(a)382 2341 y(v)m(ertex)i Fk(y)d Fi(2)d Fk(U)878 2355 y Fh(2)938 2341 y Fi(n)c Fk(U)1066 2355 y Fh(1)1136 2341 y Fp(suc)m(h)30 b(that)h Fk(x)25 b(<)g(y)s Fp(.)523 2454 y(By)i(means)g(of)h (considered)d(ab)s(o)m(v)m(e)k(F)-8 b(ord)27 b(and)g(F)-8 b(ulk)m(erson's)26 b(metho)s(dology)i(suc)m(h)e(a)382 2567 y(maxim)m(um)32 b(an)m(tipath)h Fi(U)43 b Fp(of)33 b(the)h(TCG)1794 2544 y Fk(~)1780 2567 y(G)1852 2581 y Fj(t)1881 2567 y Fp(\()p Fk(X)1998 2534 y Fh(0)2038 2567 y Fp(\))g(can)g(b)s(e)f(found)f(that)i(precedes)f(an)m(y)382 2680 y(other)h(maxim)m(um)f(an)m(tipath)h Fk(U)1483 2694 y Fh(1)1523 2680 y Fp(,)h(i.e.)52 b Fi(U)1808 2694 y Fh(1)1879 2680 y Fi(\036)31 b(U)43 b Fp(for)34 b(an)m(y)h(an)m(tipath)f Fi(U)2827 2694 y Fh(1)2900 2680 y Fp(of)h(the)f(TCG)395 2770 y Fk(~)382 2793 y(G)454 2807 y Fj(t)483 2793 y Fp(\()p Fk(X)600 2760 y Fh(0)640 2793 y Fp(\).)41 b(This)29 b(an)m(tipath)h(of) h(the)f(graph)1847 2770 y Fk(~)1833 2793 y(G)1905 2807 y Fj(t)1934 2793 y Fp(\()p Fk(X)2051 2760 y Fh(0)2091 2793 y Fp(\))h(w)m(e)g(will)c(call)j Fg(gener)-5 b(al)p Fp(.)523 2906 y(In)39 b(addition)e(to)k(the)e(general)h(an)m(tipath,)i (w)m(e)d(ma)m(y)h(\014nd)e(other)i(maxim)m(um)e(an-)382 3019 y(tipaths)44 b(of)836 2996 y Fk(~)823 3019 y(G)895 3033 y Fj(t)924 3019 y Fp(\()p Fk(X)1041 2986 y Fh(0)1081 3019 y Fp(\))h(if)f(they)h(exist.)83 b(So)45 b(for)f(an)m(y)h(v)m (ertex)h Fk(x)j Fi(2)g Fk(X)j Fp(of)45 b(the)g(TCG)395 3109 y Fk(~)382 3132 y(G)454 3146 y Fj(t)483 3132 y Fp(\()p Fk(X)600 3099 y Fh(0)640 3132 y Fp(\),)28 b(it)f(is)f(p)s(ossible)e(to) k(\014nd)d(a)i(maxim)m(um)f(an)m(tipath)g Fi(U)9 b Fp(\()p Fk(x)p Fp(\))28 b(suc)m(h)e(that)i Fk(x)d Fi(2)g(U)9 b Fp(\()p Fk(x)p Fp(\).)382 3245 y(T)-8 b(ec)m(hnically)g(,)27 b(to)g(\014nd)d(the)j(an)m(tipath)f Fi(U)9 b Fp(\()p Fk(x)p Fp(\),)28 b(it)d(is)h(su\016cien)m(t,)g(in)f(the)i(adjacen)m(t)g (matrix)382 3357 y(of)505 3334 y Fk(~)492 3357 y(G)564 3371 y Fj(t)593 3357 y Fp(\()p Fk(X)710 3324 y Fh(0)750 3357 y Fp(\))37 b(con)m(taining)g(the)g(maxim)m(um)e(n)m(um)m(b)s(er)h (of)h(units)e(in)h(allo)m(w)m(able)g(cells,)i(to)382 3470 y(mark)c(a)g(ro)m(w,)h(corresp)s(onding)d(to)j(v)m(ertex)g Fk(x)p Fp(,)g(and)e(to)i(execute)g(a)g(cycle)f(of)g(app)s(oin)m(t-)382 3583 y(men)m(t)e(of)f(marks.)44 b(Clearly)-8 b(,)31 b(the)h(an)m (tipath)f Fi(U)9 b Fp(\()p Fk(x)p Fp(\))32 b(will)d(b)s(e)i(general)g (for)g(the)h(v)m(ertex)g Fk(x)p Fp(,)382 3696 y(that)f(is,)f(an)m(y)g (other)h(an)m(tipath,)f(con)m(taining)g(v)m(ertex)i Fk(x)p Fp(,)e(will)e(precede)i(this)g(an)m(tipath.)523 3809 y(Notice)d(that)f(in)e(general)i(case)g(it)f(is)g(not)h(true)f(that)i (for)e(an)m(y)h(v)m(ertex)h Fk(x)e Fi(2)g Fk(X)33 b Fp(of)25 b(the)382 3922 y(TCG)624 3899 y Fk(~)610 3922 y(G)682 3936 y Fj(t)712 3922 y Fp(\()p Fk(X)829 3889 y Fh(0)869 3922 y Fp(\))h(there)g(exists)g(a)g(maxim)m(um)f(an)m(tipath)g Fi(U)9 b Fp(\()p Fk(x)p Fp(\).)40 b(F)-8 b(or)27 b(example,)g(consider) 382 4035 y(the)34 b(digraph)e(of)i(the)h(partially)d(ordered)h(set,)j (sho)m(wn)d(in)g(Fig.)52 b(1.)g(It)34 b(is)f(easy)h(to)h(see)382 4148 y(that)c(there)f(are)h(no)f(maxim)m(um)g(an)m(tipaths)g Fi(U)9 b Fp(\()p Fk(a)2065 4162 y Fh(1)2105 4148 y Fp(\))30 b(and)g Fi(U)9 b Fp(\()p Fk(a)2496 4162 y Fh(2)2536 4148 y Fp(\))30 b(in)f(it.)523 4261 y(A)d(directed)e(subgraph)1364 4238 y Fk(~)1350 4261 y(G)o Fp(\()p Fk(X)1538 4228 y Fj(k)1582 4261 y Fp(\))i(\()p Fk(k)i Fp(=)p 1849 4192 323 4 v 25 w(0)p Fk(;)15 b(m)21 b Fi(\000)f Fp(1)q(\))25 b(of)h(the)f(digraph)2825 4238 y Fk(~)2811 4261 y(G)p Fp(\()p Fk(X)3000 4228 y Fh(0)3040 4261 y Fp(\))g(w)m(e)h(will)382 4374 y(call)i Fg(vertex-satur)-5 b(ate)g(d)32 b(with)g(r)-5 b(esp)g(e)g(ct)32 b(to)g(the)f(initiating)g(set)g Fk(X)2564 4341 y Fj(k)2607 4374 y Fp(,)e(if,)f(in)e(its)i(transitiv)m(e)382 4487 y(closure)37 b(graph)971 4464 y Fk(~)958 4487 y(G)1030 4501 y Fj(t)1059 4487 y Fp(\()p Fk(X)1176 4454 y Fj(k)1219 4487 y Fp(\),)j(an)m(y)e(maxim)m(um)e(an)m(tipath)h Fi(U)9 b Fp(\()p Fk(x)p Fp(\))38 b Fi(\032)f Fk(X)7 b Fp(,)39 b(when)e(it)g(exists,)382 4599 y(is)d(a)i(MIS)f(of)h(the)g(subgraph) 1445 4576 y Fk(~)1431 4599 y(G)p Fp(\()p Fk(X)1620 4566 y Fj(k)1663 4599 y Fp(\))g(and)f(satis\014es)g(the)g(relation:)51 b Fk(C)7 b(ar)s(d)p Fp(\()p Fi(U)i Fp(\()p Fk(x)p Fp(\)\))34 b(=)382 4712 y Fk(C)7 b(ar)s(d)p Fp(\()p Fk(X)710 4679 y Fj(k)753 4712 y Fp(\).)41 b(Eviden)m(tly)-8 b(,)30 b(the)h(subgraph)1850 4689 y Fk(~)1836 4712 y(G)p Fp(\()p Fk(X)2025 4679 y Fj(m)2092 4712 y Fp(\))g(has)f(no)h(arcs,)g(and)f (therefore)h(it)f(is)382 4825 y(v)m(ertex-saturated)i(with)d(resp)s (ect)i(to)g(its)f(initiating)d(set)k(alw)m(a)m(ys.)523 4938 y(A)38 b(CCD)881 4915 y Fk(~)867 4938 y(G)p Fp(\()p Fk(X)1056 4905 y Fh(0)1096 4938 y Fp(\))g(is)f(called)g Fg(vertex-satur)-5 b(ate)g(d)41 b(\(VS-digr)-5 b(aph\))40 b Fp(if)d(an)m(y)h(of)f(its)h(di-)382 5051 y(rected)j(subgraphs)1119 5028 y Fk(~)1105 5051 y(G)p Fp(\()p Fk(X)1294 5018 y Fj(k)1337 5051 y Fp(\))f(\()p Fk(k)46 b Fp(=)p 1652 4983 166 4 v 41 w(0)p Fk(;)15 b(m)q Fp(\),)43 b(induced)38 b(b)m(y)i(the)g(la)m(y)m(er)h Fk(X)2883 5018 y Fj(k)2926 5051 y Fp(,)i(is)c(v)m(ertex-)382 5164 y(saturated)31 b(with)e(resp)s(ect)h(to)h(the)g(initiating)d(set)j Fk(X)2191 5131 y Fj(k)2234 5164 y Fp(.)523 5277 y(Notice)g(that)g(digraph,)e (represen)m(ted)i(in)e(Fig.)40 b(2,)31 b(is)f(not)g(v)m (ertex-saturated.)1831 5652 y(11)p eop %%Page: 12 12 12 11 bop 523 548 a Fp(Let)35 b(there)g(b)s(e)g(some)g(digraph)1640 525 y Fk(~)1626 548 y(G)p Fp(\()p Fk(X)1815 515 y Fh(0)1855 548 y Fp(\).)54 b(T)-8 b(o)36 b(construct)f(a)g(VS-digraph,)g(w)m(e)g (use)382 661 y(the)30 b(follo)m(wing)f(algorithm.)382 845 y Fg(Step)k(1.)46 b Fp(Put)30 b Fk(k)e Fp(:=)e(0)k(and)g Fk(\013)c Fp(:=)k Fq(false)p Fp(.)382 981 y Fg(Step)j(2.)46 b Fp(Find)29 b(the)h(transitiv)m(e)g(closure)g(graph)2056 958 y Fk(~)2043 981 y(G)2115 995 y Fj(t)2144 981 y Fp(\()p Fk(X)2261 948 y Fj(k)2304 981 y Fp(\).)382 1118 y Fg(Step)j(3.)46 b Fp(Construct)30 b(an)g(MPP)g(of)1610 1095 y Fk(~)1596 1118 y(G)1668 1132 y Fj(t)1697 1118 y Fp(\()p Fk(X)1814 1085 y Fj(k)1858 1118 y Fp(\).)382 1254 y Fg(Step)j(4.)46 b Fp(Find)35 b(the)i(maxim)m(um)f(an)m(tipath)h(of)2017 1231 y Fk(~)2003 1254 y(G)2075 1268 y Fj(t)2104 1254 y Fp(\()p Fk(X)2221 1221 y Fj(k)2265 1254 y Fp(\))g(for)f(eac)m(h)i(v)m (ertex)h Fk(x)d Fi(2)f Fk(X)3242 1221 y Fj(k)3310 1254 y Fi([)609 1367 y(\001)15 b(\001)g(\001)21 b([)f Fk(X)898 1334 y Fj(m)965 1367 y Fp(.)382 1504 y Fg(Step)33 b(5.)46 b Fp(Chec)m(k)36 b(whether)e(eac)m(h)j(of)f(the)f(found)f(maxim)m(um)h (an)m(tipaths)g Fi(U)9 b Fp(\()p Fk(x)p Fp(\))36 b(of)f(the)609 1617 y(graph)896 1594 y Fk(~)882 1617 y(G)954 1631 y Fj(t)983 1617 y Fp(\()p Fk(X)1100 1584 y Fj(k)1144 1617 y Fp(\))46 b(is)f(a)h(MIS)f(of)h(the)g(digraph)2288 1594 y Fk(~)2275 1617 y(G)o Fp(\()p Fk(X)2463 1584 y Fj(k)2506 1617 y Fp(\))g(and)g Fk(C)7 b(ar)s(d)p Fp(\()p Fi(U)i Fp(\()p Fk(x)p Fp(\)\))51 b(=)609 1730 y Fk(C)7 b(ar)s(d)p Fp(\()p Fk(X)937 1697 y Fj(k)980 1730 y Fp(\).)40 b(If)25 b(it)h(is)g(true,)h(\014nish)d(the)i(design)f(of)i(the)f(digraph)2815 1707 y Fk(~)2802 1730 y(G)o Fp(\()p Fk(X)2990 1697 y Fj(k)3034 1730 y Fp(\))g(v)m(ertex-)609 1842 y(saturated)31 b(with)e(resp)s(ect)h(to)i(the)e(initiating)e(set)j Fk(X)2418 1809 y Fj(k)2461 1842 y Fp(.)41 b(Go)31 b(to)g(Step)f(6.)609 1979 y(Otherwise)d(complete)h(the)h(found)d(an)m(tipath)i Fi(U)9 b Fp(\()p Fk(x)p Fp(\))29 b(\(when)e(it)h(is)f(necessary\))i(to) 609 2092 y(a)f(MIS,)g(put)f Fk(W)38 b Fp(:=)25 b Fi(U)9 b Fp(\()p Fk(x)p Fp(\))28 b(and)f(construct)i(a)f(new)f(acyclic)h (digraph)2993 2069 y Fk(~)2980 2092 y(G)o Fp(\()p Fk(X)3168 2059 y Fj(k)3212 2092 y Fp(\))g(b)m(y)609 2205 y(the)36 b(cutting)g(op)s(eration)g Fk(\033)1555 2219 y Fj(W)1635 2205 y Fp(\()1684 2182 y Fk(~)1670 2205 y(G)p Fp(\()p Fk(X)1859 2172 y Fh(0)1899 2205 y Fp(\)\),)j(put)c Fk(\013)h Fp(:=)g Fq(true)f Fp(and)h(construct)g(the)609 2318 y(new)30 b(digraph)1143 2295 y Fk(~)1130 2318 y(G)o Fp(\()p Fk(X)1318 2285 y Fh(0)1358 2318 y Fp(\).)41 b(Return)30 b(to)h(Step)f(2.)382 2454 y Fg(Step)j(6.)46 b Fp(Compute)d Fk(k)52 b Fp(:=)c Fk(k)32 b Fp(+)d(1.)82 b(If)44 b Fk(k)51 b(<)d(m)c Fp(then)g (distinguish)c(the)k(transitiv)m(e)609 2567 y(closure)34 b(graph)1193 2544 y Fk(~)1180 2567 y(G)1252 2581 y Fj(t)1281 2567 y Fp(\()p Fk(X)1398 2534 y Fj(k)1441 2567 y Fp(\))h(from)1744 2544 y Fk(~)1731 2567 y(G)1803 2581 y Fj(t)1832 2567 y Fp(\()p Fk(X)1949 2534 y Fj(k)r Fv(\000)p Fh(1)2082 2567 y Fp(\),)i(k)m(eeping)e(all)e(paths)i(of)g(the)g(MPP)609 2680 y(of)732 2657 y Fk(~)718 2680 y(G)790 2694 y Fj(t)819 2680 y Fp(\()p Fk(X)936 2647 y Fj(k)r Fv(\000)p Fh(1)1070 2680 y Fp(\))h(whic)m(h)e(are)i(inciden)m(t)f(to)h(v)m(ertices)g(of)g (the)g(new)f(graph.)56 b(Go)37 b(to)609 2793 y(Step)30 b(4.)609 2930 y(If)i Fk(k)e Fp(=)e Fk(m)j Fp(and)g Fk(\013)e Fp(=)i Fq(true)p Fp(,)h(go)h(to)f(Step)g(1.)45 b(If)31 b Fk(k)g Fp(=)d Fk(m)j Fp(and)g Fk(\013)e Fp(=)i Fq(false)p Fp(,)h(go)h(to)609 3042 y(Step)d(7.)382 3179 y Fg(Step)j(7.)46 b Fp(T)-8 b(erminate.)40 b(A)31 b(VS-digraph)d(is)i(constructed.)382 3384 y Fq(Theorem)k(3)46 b Fg(VS-digr)-5 b(aph)34 b(c)-5 b(an)33 b(b)-5 b(e)32 b(c)-5 b(onstructe)g(d)35 b(in)d(time)h Fk(O)s Fp(\()p Fk(n)2633 3351 y Fh(5)2672 3384 y Fp(\))p Fg(.)523 3590 y Fp(One)g(completion)g(of)h(the)g(steps)f(1)h({)g(6)g (requires)f Fk(O)s Fp(\()p Fk(n)2429 3557 y Fh(3)2429 3616 y Fj(k)2471 3590 y Fp(\))h(time)f(units,)g(where)g Fk(n)3328 3605 y Fj(k)382 3712 y Fp(is)g(the)i(size)f(of)h(the)g(v)m (ertex)g(set)h(of)1630 3689 y Fk(~)1616 3712 y(G)p Fp(\()p Fk(X)1805 3679 y Fj(k)1848 3712 y Fp(\).)54 b(Assuming)32 b(that)j(for)g(eac)m(h)g(completion)382 3825 y(of)42 b(these)h(steps)f(the)g(maxim)m(um)f(an)m(tipath)h(increases)g(at)h (one)f(v)m(ertex,)47 b(w)m(e)42 b(obtain)382 3938 y(the)e(design)e (time)h(of)h(a)g(digraph)1605 3915 y Fk(~)1591 3938 y(G)p Fp(\()p Fk(X)1780 3905 y Fj(k)1823 3938 y Fp(\))g(v)m(ertex-saturated)i (with)c(resp)s(ect)h(to)h(the)382 4051 y(initiating)28 b(set,)j(is)e(equal)h(to)h Fk(O)s Fp(\()p Fk(n)1547 4018 y Fh(4)1547 4077 y Fj(k)1590 4051 y Fp(\).)523 4164 y(Hence,)42 b(the)d(design)e(time)i(of)g(a)g(correctly)g(constructed)g(v)m (ertex-saturated)i(di-)382 4277 y(graph)653 4254 y Fk(~)640 4277 y(G)o Fp(\()p Fk(X)828 4244 y Fh(0)868 4277 y Fp(\))31 b(is)f(equal)f(to:)1497 4393 y Ff(X)1497 4578 y Fv(8)p Fj(n)1580 4590 y Fb(k)1633 4474 y Fk(O)s Fp(\()p Fk(n)1795 4436 y Fh(4)1795 4496 y Fj(k)1837 4474 y Fp(\))d(=)f Fk(O)s Fp(\()p Fk(n)2156 4436 y Fh(5)2195 4474 y Fp(\))p Fk(:)3093 4752 y Fp(Q.E.D.)523 4965 y Fq(Example)34 b(1.)523 5177 y Fp(Fig.)k(3)24 b(sho)m(ws)g(a)g(digraph)1449 5154 y Fk(~)1436 5177 y(G)o Fp(\()p Fk(X)1624 5144 y Fh(0)1664 5177 y Fp(\),)i(obtained)d(from)g(an)g(initial)e(undirected)h(graph)382 5290 y Fk(G)e Fp(as)h(it)f(w)m(as)h(describ)s(ed)e(ab)s(o)m(v)m(e.)38 b(Recall)20 b(that)i(the)e(orien)m(tation)h(of)g(arcs)g(of)g(the)f (digraph)382 5403 y(in)29 b(\014gures)h(is)f(from)h(b)s(elo)m(w)f(up)m (w)m(ards.)1831 5652 y(12)p eop %%Page: 13 13 13 12 bop 523 548 a Fp(Construct)30 b(a)h(VS-digraph.)523 661 y(The)h(adjacen)m(t)i(matrix)e(of)g(the)h(TCG)1888 638 y Fk(~)1875 661 y(G)1947 675 y Fj(t)1976 661 y Fp(\()p Fk(X)2093 628 y Fh(0)2133 661 y Fp(\))g(is)e(sho)m(wn)h(in)g(Fig.)47 b(4.)g(The)32 b(\014cti-)382 774 y(tious)h(arcs)h(of)g(this)f(graph)g (are)i(represen)m(ted)f(b)m(y)f(the)h(letter)h Fq(f)p Fp(.)51 b(Arcs,)35 b(b)s(elonging)d(to)382 887 y(the)d(MPP)g(of)888 864 y Fk(~)875 887 y(G)947 901 y Fj(t)976 887 y Fp(\()p Fk(X)1093 854 y Fh(0)1133 887 y Fp(\),)h(are)f(put)f(in)m(to)h (circles.)39 b(These)29 b(arcs)g(are)g(sho)m(wn)f(in)f(Fig.)41 b(3)29 b(b)m(y)382 1000 y(thic)m(k)h(lines.)523 1112 y(First)h(of)h(all,)f(notice)i(that)f(the)g(initiating)d(set)k Fk(X)2251 1080 y Fh(0)2322 1112 y Fp(=)f Fi(f)p Fk(x)2522 1126 y Fh(1)2561 1112 y Fk(;)15 b(x)2653 1126 y Fh(2)2693 1112 y Fk(;)g(x)2785 1126 y Fh(3)2825 1112 y Fk(;)g(x)2917 1126 y Fh(4)2957 1112 y Fi(g)32 b Fp(is)f(a)h(MIS)382 1225 y(of)499 1202 y Fk(~)485 1225 y(G)p Fp(\()p Fk(X)674 1192 y Fh(0)714 1225 y Fp(\).)523 1338 y(Find)d(the)i(general)f(an)m (tipath)g Fi(U)40 b Fp(\(see)31 b(Fig.)41 b(4)31 b(\(a\)\).)42 b(The)30 b(set)h(of)g(mark)m(ed)f(ro)m(ws)h(is)382 1451 y Fk(U)444 1465 y Fj(r)512 1451 y Fp(=)e Fi(f)p Fp(5)p Fk(;)15 b Fp(7)p Fk(;)g Fp(8)p Fk(;)g Fp(10)p Fi(g)p Fp(,)34 b(and)29 b(the)h(set)h(of)e(mark)m(ed)h(columns)f(is)f(empt)m (y)-8 b(,)31 b(that)g(is,)e Fk(U)3119 1465 y Fj(c)3179 1451 y Fp(=)c Fi(\013)p Fp(.)382 1564 y(Therefore:)990 1677 y Fi(U)34 b Fp(=)25 b Fk(U)1239 1691 y Fj(r)1297 1677 y Fi(n)c Fk(U)1425 1691 y Fj(c)1485 1677 y Fp(=)k Fi(f)p Fp(5)p Fk(;)15 b Fp(7)p Fk(;)g Fp(8)p Fk(;)g Fp(10)p Fi(g)24 b(n)d(\013)k Fp(=)g Fi(f)p Fp(5)p Fk(;)15 b Fp(7)p Fk(;)g Fp(8)p Fk(;)g Fp(10)p Fi(g)p Fk(:)523 1844 y Fp(This)29 b(an)m(tipath)h(is)f(a)i(MIS)f(of)g(the)h(digraph,)e(and)h Fk(C)7 b(ar)s(d)p Fp(\()p Fi(U)i Fp(\))30 b(=)g Fk(C)7 b(ar)s(d)p Fp(\()p Fk(X)3063 1811 y Fh(0)3103 1844 y Fp(\).)523 1957 y(Find)29 b(the)i(maxim)m(um)e(an)m(tipaths)h Fi(U)9 b Fp(\()p Fk(x)p Fp(\))31 b(for)f(graph)g(v)m(ertices.)523 2070 y(Clearly)-8 b(,)30 b Fi(U)9 b Fp(\()p Fk(x)1010 2084 y Fh(5)1049 2070 y Fp(\))31 b(=)f Fi(U)9 b Fp(\()p Fk(x)1369 2084 y Fh(7)1409 2070 y Fp(\))30 b(=)g Fi(U)9 b Fp(\()p Fk(x)1728 2084 y Fh(8)1768 2070 y Fp(\))31 b(=)f Fi(U)9 b Fp(\()p Fk(x)2088 2084 y Fh(10)2163 2070 y Fp(\))30 b(=)g Fi(U)9 b Fp(.)523 2183 y(T)-8 b(o)31 b(\014nd)e Fi(U)9 b Fp(\()p Fk(x)992 2197 y Fh(1)1031 2183 y Fp(\),)31 b(mark)f(the)h(\014rst)f(ro)m(w)g(of)h(the)f(adjacen)m (t)i(matrix)e(of)2913 2160 y Fk(~)2899 2183 y(G)2971 2197 y Fj(t)3000 2183 y Fp(\()p Fk(X)3117 2150 y Fh(0)3157 2183 y Fp(\).)523 2296 y(Marking)d(the)i(\014rst)e(ro)m(w)h(in)e(Fig.) 40 b(4)28 b(\(b\),)h(w)m(e)g(ha)m(v)m(e)g Fk(U)2343 2310 y Fj(r)2409 2296 y Fp(=)e Fi(f)p Fp(1)p Fk(;)15 b Fp(3)p Fk(;)g Fp(4)p Fk(;)g Fp(5)p Fk(;)g Fp(6)p Fk(;)g Fp(7)p Fk(;)g Fp(8)p Fk(:)p Fp(9)q Fk(:)p Fp(1)q(0)p Fi(g)p Fp(,)382 2408 y(and)30 b Fk(U)621 2422 y Fj(c)686 2408 y Fp(=)g Fi(f)p Fp(5)p Fk(;)15 b Fp(7)p Fk(;)g Fp(8)p Fk(;)g Fp(9)p Fk(;)g Fp(10)p Fi(g)p Fp(.)46 b(Consequen)m(tly)-8 b(,)1290 2613 y Fi(U)9 b Fp(\()p Fk(x)1443 2627 y Fh(1)1483 2613 y Fp(\))25 b(=)g Fk(U)1701 2627 y Fj(r)1760 2613 y Fi(n)20 b Fk(U)1887 2627 y Fj(c)1947 2613 y Fp(=)25 b Fi(f)p Fp(1)p Fk(;)15 b Fp(3)p Fk(;)g Fp(4)p Fk(;)g Fp(6)p Fi(g)p Fk(:)523 2817 y Fp(This)29 b(an)m(tipath)h(is)f(a)i(MIS)f (of)g(the)h(digraph,)e(and)h Fk(C)7 b(ar)s(d)p Fp(\()p Fi(U)i Fp(\()p Fk(x)2656 2831 y Fh(1)2695 2817 y Fp(\)\))31 b(=)f Fk(C)7 b(ar)s(d)p Fp(\()p Fk(X)3225 2784 y Fh(0)3265 2817 y Fp(\).)523 2930 y(Similarly)-8 b(,)27 b(marking)j(the)g(second)h (ro)m(w)f(in)f(Fig.)41 b(4)31 b(\(c\),)g(w)m(e)g(ha)m(v)m(e)606 3134 y Fi(U)9 b Fp(\()p Fk(x)759 3148 y Fh(2)798 3134 y Fp(\))26 b(=)f Fi(f)p Fp(1)p Fk(;)15 b Fp(2)p Fk(;)g Fp(3)p Fk(;)g Fp(4)p Fk(;)g Fp(5)p Fk(;)g Fp(6)p Fk(;)g Fp(7)p Fk(;)g Fp(8)q Fk(;)g Fp(9)q Fk(;)g Fp(1)q(0)p Fi(g)27 b(n)20 b(f)p Fp(5)p Fk(;)15 b Fp(6)p Fk(;)g Fp(7)p Fk(;)g Fp(8)p Fk(;)g Fp(9)p Fk(;)g Fp(10)p Fi(g)32 b Fp(=)25 b Fi(f)p Fp(1)p Fk(;)15 b Fp(2)p Fk(;)g Fp(3)p Fk(;)g Fp(4)p Fi(g)p Fk(:)523 3338 y Fp(This)28 b(maxim)m(um)i(an)m (tipath)g(is)f(also)h(a)h(MIS)e(of)i(the)f(digraph,)f(and)h Fk(C)7 b(ar)s(d)p Fp(\()p Fi(U)i Fp(\()p Fk(x)3261 3352 y Fh(2)3300 3338 y Fp(\)\))382 3451 y(=)30 b Fk(C)7 b(ar)s(d)p Fp(\()p Fk(X)811 3418 y Fh(0)851 3451 y Fp(\).)523 3564 y(Similarly)-8 b(,)27 b(w)m(e)k(obtain)f(the)h(maxim)m(um)e(an)m (tipaths)1492 3768 y Fi(U)9 b Fp(\()p Fk(x)1645 3782 y Fh(3)1685 3768 y Fp(\))25 b(=)g Fi(f)p Fp(3)p Fk(;)15 b Fp(4)p Fk(;)g Fp(5)p Fk(;)g Fp(8)p Fi(g)p Fk(;)1492 3973 y Fi(U)9 b Fp(\()p Fk(x)1645 3987 y Fh(4)1685 3973 y Fp(\))25 b(=)g Fi(f)p Fp(4)p Fk(;)15 b Fp(5)p Fk(;)g Fp(8)p Fk(;)g Fp(9)p Fi(g)p Fk(;)1311 5083 y 3.3 setlinewidth 1490 4811 a Fc(@)1573 4894 y(@)1643 4964 y(@)1685 4575 y(\001)1643 4658 y(\001)1602 4741 y(\001)1560 4824 y(\001)1519 4907 y(\001)1490 4964 y(\001)1726 4728 y(\023)1788 4645 y(\023)1851 4562 y(\023)1913 4479 y(\023)1975 4396 y(\023)2018 4339 y(\023)-38 b(\001)2000 4422 y(\001)1959 4505 y(\001)1917 4588 y(\001)1876 4671 y(\001)1834 4754 y(\001)1793 4837 y(\001)1751 4920 y(\001)1729 4964 y(\001)1726 4965 y(\023)1788 4881 y(\023)1851 4798 y(\023)1913 4715 y(\023)1975 4632 y(\023)2018 4575 y(\023)2000 4547 y(\021)1917 4603 y(\021)1834 4658 y(\021)1751 4713 y(\021)1729 4728 y(\021)2080 4965 y(\000)2163 4881 y(\000)2234 4811 y(\000)2289 4988 y(B)2261 4905 y(B)2234 4822 y(B)2206 4739 y(B)2178 4656 y(B)2151 4573 y(B)2123 4490 y(B)2095 4407 y(B)2071 4334 y(B)1490 4965 y Fd(x)161 b(x)279 b(x)162 b(x)1490 4728 y(x)f(x)516 b(x)1726 4492 y(x)279 b(x)2080 4256 y(x)1515 5060 y Fk(x)1567 5074 y Fh(1)1751 5060 y Fk(x)1803 5074 y Fh(2)2106 5060 y Fk(x)2158 5074 y Fh(3)2365 5060 y Fk(x)2417 5074 y Fh(4)1397 4647 y Fk(x)1449 4661 y Fh(5)1610 4788 y Fk(x)1662 4802 y Fh(6)2342 4647 y Fk(x)2394 4661 y Fh(7)1610 4410 y Fk(x)1662 4424 y Fh(8)2153 4410 y Fk(x)2205 4424 y Fh(9)2064 4174 y Fk(x)2116 4188 y Fh(10)p 1477 4965 25 237 v 1714 4965 25 473 v 2068 4965 25 709 v 2304 4988 25 260 v 1527 5279 a Fp(Fig.)31 b(3:)41 b(A)30 b(digraph)1831 5652 y(13)p eop %%Page: 14 14 14 13 bop 384 5064 a 3.3 setlinewidth 561 3340 4 1182 v 563 2160 1182 4 v 1742 3340 4 1182 v 563 3341 1182 4 v 563 2278 V 563 2397 V 563 2515 V 563 2633 V 563 2751 V 563 2869 V 563 2987 V 563 3105 V 563 3223 V 679 3340 4 1182 v 797 3340 V 915 3340 V 1033 3340 V 1152 3340 V 1270 3340 V 1388 3340 V 1506 3340 V 1624 3340 V 423 2230 a Fk(x)475 2244 y Fh(1)423 2349 y Fk(x)475 2363 y Fh(2)423 2467 y Fk(x)475 2481 y Fh(3)423 2585 y Fk(x)475 2599 y Fh(4)423 2703 y Fk(x)475 2717 y Fh(5)423 2821 y Fk(x)475 2835 y Fh(6)423 2939 y Fk(x)475 2953 y Fh(7)423 3057 y Fk(x)475 3071 y Fh(8)423 3175 y Fk(x)475 3189 y Fh(9)405 3293 y Fk(x)457 3307 y Fh(10)564 2106 y Fk(x)616 2120 y Fh(1)682 2106 y Fk(x)734 2120 y Fh(2)800 2106 y Fk(x)852 2120 y Fh(3)919 2106 y Fk(x)971 2120 y Fh(4)1037 2106 y Fk(x)1089 2120 y Fh(5)1155 2106 y Fk(x)1207 2120 y Fh(6)1273 2106 y Fk(x)1325 2120 y Fh(7)1391 2106 y Fk(x)1443 2120 y Fh(8)1509 2106 y Fk(x)1561 2120 y Fh(9)1610 2106 y Fk(x)1662 2120 y Fh(10)1071 2247 y Fp(1)1094 2217 y Fd(j)1426 2247 y Fp(1)1071 2365 y(1)74 b(1)1212 2335 y Fd(j)1435 2367 y Fp(f)1544 2365 y(1)f(1)1308 2483 y(1)191 b(1)1567 2453 y Fd(j)1671 2486 y Fp(f)1308 2601 y(1)1330 2572 y Fd(j)1662 2601 y Fp(1)1426 2838 y(1)1449 2808 y Fd(j)1544 2838 y Fp(1)73 b(1)1662 3192 y(1)1685 3162 y Fd(j)1780 2247 y Fp(5)1780 2357 y Fi(\003)1780 2483 y Fp(9)1780 2601 y(7)1780 2711 y Fi(\003)1780 2838 y Fp(8)1780 2948 y Fi(\003)1780 3066 y(\003)1757 3192 y Fp(10)1780 3302 y Fi(\003)1662 3428 y Fp(2)-163 b(2)g(2)g(3)g(2)f(2) 1039 3540 y(\(c\))p 2096 3340 V 2098 2160 1182 4 v 3278 3340 4 1182 v 2098 3341 1182 4 v 2098 2278 V 2098 2397 V 2098 2515 V 2098 2633 V 2098 2751 V 2098 2869 V 2098 2987 V 2098 3105 V 2098 3223 V 2215 3340 4 1182 v 2333 3340 V 2451 3340 V 2569 3340 V 2687 3340 V 2805 3340 V 2923 3340 V 3041 3340 V 3159 3340 V 1958 2230 a Fk(x)2010 2244 y Fh(1)1958 2349 y Fk(x)2010 2363 y Fh(2)1958 2467 y Fk(x)2010 2481 y Fh(3)1958 2585 y Fk(x)2010 2599 y Fh(4)1958 2703 y Fk(x)2010 2717 y Fh(5)1958 2821 y Fk(x)2010 2835 y Fh(6)1958 2939 y Fk(x)2010 2953 y Fh(7)1958 3057 y Fk(x)2010 3071 y Fh(8)1958 3175 y Fk(x)2010 3189 y Fh(9)1940 3293 y Fk(x)1992 3307 y Fh(10)2100 2106 y Fk(x)2152 2120 y Fh(1)2218 2106 y Fk(x)2270 2120 y Fh(2)2336 2106 y Fk(x)2388 2120 y Fh(3)2454 2106 y Fk(x)2506 2120 y Fh(4)2572 2106 y Fk(x)2624 2120 y Fh(5)2690 2106 y Fk(x)2742 2120 y Fh(6)2808 2106 y Fk(x)2860 2120 y Fh(7)2926 2106 y Fk(x)2978 2120 y Fh(8)3045 2106 y Fk(x)3097 2120 y Fh(9)3145 2106 y Fk(x)3197 2120 y Fh(10)2607 2247 y Fp(1)2630 2217 y Fd(j)2961 2247 y Fp(1)2607 2365 y(1)73 b(1)2748 2335 y Fd(j)2970 2367 y Fp(f)2961 2365 y(1)191 b(1)2843 2483 y(1)g(1)3102 2453 y Fd(j)3206 2486 y Fp(f)2843 2601 y(1)2866 2572 y Fd(j)3197 2601 y Fp(1)2961 2838 y(1)2984 2808 y Fd(j)3079 2838 y Fp(1)73 b(1)3197 3192 y(1)3220 3162 y Fd(j)3316 2475 y Fi(\003)3316 2601 y Fp(7)3316 2711 y Fi(\003)3316 2948 y(\003)3316 3066 y(\003)3293 3192 y Fp(10)3316 3302 y Fi(\003)3197 3428 y Fp(3)-163 b(3)-281 b(3)2569 3540 y(\(d\))p 561 1828 V 563 649 1182 4 v 1742 1828 4 1182 v 563 1830 1182 4 v 563 767 V 563 885 V 563 1003 V 563 1121 V 563 1239 V 563 1357 V 563 1475 V 563 1593 V 563 1712 V 679 1828 4 1182 v 797 1828 V 915 1828 V 1033 1828 V 1152 1828 V 1270 1828 V 1388 1828 V 1506 1828 V 1624 1828 V 423 719 a Fk(x)475 733 y Fh(1)423 837 y Fk(x)475 851 y Fh(2)423 955 y Fk(x)475 969 y Fh(3)423 1073 y Fk(x)475 1087 y Fh(4)423 955 y Fk(x)475 969 y Fh(5)423 1309 y Fk(x)475 1323 y Fh(6)423 1427 y Fk(x)475 1441 y Fh(7)423 1545 y Fk(x)475 1559 y Fh(8)423 1664 y Fk(x)475 1678 y Fh(9)405 1782 y Fk(x)457 1796 y Fh(10)564 595 y Fk(x)616 609 y Fh(1)682 595 y Fk(x)734 609 y Fh(2)800 595 y Fk(x)852 609 y Fh(3)919 595 y Fk(x)971 609 y Fh(4)1037 595 y Fk(x)1089 609 y Fh(5)1155 595 y Fk(x)1207 609 y Fh(6)1273 595 y Fk(x)1325 609 y Fh(7)1391 595 y Fk(x)1443 609 y Fh(8)1509 595 y Fk(x)1561 609 y Fh(9)1610 595 y Fk(x)1662 609 y Fh(10)1071 735 y Fp(1)1094 705 y Fd(j)1426 735 y Fp(1)1071 853 y(1)74 b(1)1212 824 y Fd(j)1435 856 y Fp(f)1544 853 y(1)f(1)1308 971 y(1)191 b(1)1567 942 y Fd(j)1671 974 y Fp(f)1308 1090 y(1)1330 1060 y Fd(j)1662 1090 y Fp(1)1426 1326 y(1)1449 1296 y Fd(j)1544 1326 y Fp(1)73 b(1)1662 1680 y(1)1685 1650 y Fd(j)1780 1200 y Fi(\003)1780 1436 y(\003)1780 1554 y(\003)1780 1790 y(\003)1036 2028 y Fp(\(a\))p 2096 1828 V 2098 649 1182 4 v 3278 1828 4 1182 v 2098 1830 1182 4 v 2098 767 V 2098 885 V 2098 1003 V 2098 1121 V 2098 1239 V 2098 1357 V 2098 1475 V 2098 1593 V 2098 1712 V 2215 1828 4 1182 v 2333 1828 V 2451 1828 V 2569 1828 V 2687 1828 V 2805 1828 V 2923 1828 V 3041 1828 V 3159 1828 V 1958 719 a Fk(x)2010 733 y Fh(1)1958 837 y Fk(x)2010 851 y Fh(2)1958 955 y Fk(x)2010 969 y Fh(3)1958 1073 y Fk(x)2010 1087 y Fh(4)1958 1191 y Fk(x)2010 1205 y Fh(5)1958 1309 y Fk(x)2010 1323 y Fh(6)1958 1427 y Fk(x)2010 1441 y Fh(7)1958 1545 y Fk(x)2010 1559 y Fh(8)1958 1664 y Fk(x)2010 1678 y Fh(9)1940 1782 y Fk(x)1992 1796 y Fh(10)2100 595 y Fk(x)2152 609 y Fh(1)2218 595 y Fk(x)2270 609 y Fh(2)2336 595 y Fk(x)2388 609 y Fh(3)2454 595 y Fk(x)2506 609 y Fh(4)2572 595 y Fk(x)2624 609 y Fh(5)2690 595 y Fk(x)2742 609 y Fh(6)2808 595 y Fk(x)2860 609 y Fh(7)2926 595 y Fk(x)2978 609 y Fh(8)3045 595 y Fk(x)3097 609 y Fh(9)3145 595 y Fk(x)3197 609 y Fh(10)2607 735 y Fp(1)2630 705 y Fd(j)2961 735 y Fp(1)2607 853 y(1)g(1)2748 824 y Fd(j)2970 856 y Fp(f)3079 853 y(1)g(1)2843 971 y(1)191 b(1)3102 942 y Fd(j)3206 974 y Fp(f)2843 1090 y(1)2866 1060 y Fd(j)3197 1090 y Fp(1)2961 1326 y(1)2984 1296 y Fd(j)3079 1326 y Fp(1)73 b(1)3197 1680 y(1)3220 1650 y Fd(j)3316 727 y Fi(\003)3316 971 y Fp(9)3316 1090 y(7)3316 1200 y Fi(\003)3316 1326 y Fp(8)3316 1436 y Fi(\003)3316 1554 y(\003)3293 1680 y Fp(10)3316 1790 y Fi(\003)3197 1916 y Fp(6)-163 b(6)g(1)g(3)-281 b(1)2569 2028 y(\(b\))p 561 4852 V 563 3672 1182 4 v 1742 4852 4 1182 v 563 4853 1182 4 v 563 3790 V 563 3908 V 563 4026 V 563 4145 V 563 4263 V 563 4381 V 563 4499 V 563 4617 V 563 4735 V 679 4852 4 1182 v 797 4852 V 915 4852 V 1033 4852 V 1152 4852 V 1270 4852 V 1388 4852 V 1506 4852 V 1624 4852 V 423 3742 a Fk(x)475 3756 y Fh(1)423 3860 y Fk(x)475 3874 y Fh(2)423 3979 y Fk(x)475 3993 y Fh(3)423 4097 y Fk(x)475 4111 y Fh(4)423 4215 y Fk(x)475 4229 y Fh(5)423 4333 y Fk(x)475 4347 y Fh(6)423 4451 y Fk(x)475 4465 y Fh(7)423 4569 y Fk(x)475 4583 y Fh(8)423 4687 y Fk(x)475 4701 y Fh(9)405 4805 y Fk(x)457 4819 y Fh(10)564 3618 y Fk(x)616 3632 y Fh(1)682 3618 y Fk(x)734 3632 y Fh(2)800 3618 y Fk(x)852 3632 y Fh(3)919 3618 y Fk(x)971 3632 y Fh(4)1037 3618 y Fk(x)1089 3632 y Fh(5)1155 3618 y Fk(x)1207 3632 y Fh(6)1273 3618 y Fk(x)1325 3632 y Fh(7)1391 3618 y Fk(x)1443 3632 y Fh(8)1509 3618 y Fk(x)1561 3632 y Fh(9)1610 3618 y Fk(x)1662 3632 y Fh(10)1071 3759 y Fp(1)1094 3729 y Fd(j)1426 3759 y Fp(1)1071 3877 y(1)74 b(1)1212 3847 y Fd(j)1435 3879 y Fp(f)1544 3877 y(1)f(1)1308 3995 y(1)191 b(1)1567 3965 y Fd(j)1671 3997 y Fp(f)1308 4113 y(1)1330 4083 y Fd(j)1662 4113 y Fp(1)1426 4349 y(1)1449 4320 y Fd(j)1544 4349 y Fp(1)73 b(1)1662 4704 y(1)1685 4674 y Fd(j)1780 4105 y Fi(\003)1780 4223 y(\003)1780 4459 y(\003)1780 4577 y(\003)1757 4704 y Fp(10)1780 4814 y Fi(\003)1662 4940 y Fp(4)-399 b(4)1039 5051 y(\(e\))p 2096 4852 V 2098 3672 1182 4 v 3278 4852 4 1182 v 2098 4853 1182 4 v 2098 3790 V 2098 3908 V 2098 4026 V 2098 4145 V 2098 4263 V 2098 4381 V 2098 4499 V 2098 4617 V 2098 4735 V 2215 4852 4 1182 v 2333 4852 V 2451 4852 V 2569 4852 V 2687 4852 V 2805 4852 V 2923 4852 V 3041 4852 V 3159 4852 V 1958 3742 a Fk(x)2010 3756 y Fh(1)1958 3860 y Fk(x)2010 3874 y Fh(2)1958 3979 y Fk(x)2010 3993 y Fh(3)1958 4097 y Fk(x)2010 4111 y Fh(4)1958 4215 y Fk(x)2010 4229 y Fh(5)1958 4333 y Fk(x)2010 4347 y Fh(6)1958 4451 y Fk(x)2010 4465 y Fh(7)1958 4569 y Fk(x)2010 4583 y Fh(8)1958 4687 y Fk(x)2010 4701 y Fh(9)1940 4805 y Fk(x)1992 4819 y Fh(10)2100 3618 y Fk(x)2152 3632 y Fh(1)2218 3618 y Fk(x)2270 3632 y Fh(2)2336 3618 y Fk(x)2388 3632 y Fh(3)2454 3618 y Fk(x)2506 3632 y Fh(4)2572 3618 y Fk(x)2624 3632 y Fh(5)2690 3618 y Fk(x)2742 3632 y Fh(6)2808 3618 y Fk(x)2860 3632 y Fh(7)2926 3618 y Fk(x)2978 3632 y Fh(8)3045 3618 y Fk(x)3097 3632 y Fh(9)3145 3618 y Fk(x)3197 3632 y Fh(10)2607 3759 y Fp(1)2630 3729 y Fd(j)2961 3759 y Fp(1)2607 3877 y(1)73 b(1)2748 3847 y Fd(j)2970 3879 y Fp(f)3079 3877 y(1)g(1)2843 3995 y(1)191 b(1)3102 3965 y Fd(j)3206 3997 y Fp(f)2843 4113 y(1)2866 4083 y Fd(j)3197 4113 y Fp(1)2961 4349 y(1)2984 4320 y Fd(j)3079 4349 y Fp(1)73 b(1)3197 4704 y(1)3220 4674 y Fd(j)3316 3995 y Fp(9)3316 4113 y(7)3316 4223 y Fi(\003)3316 4341 y(\003)3316 4459 y(\003)3316 4577 y(\003)3293 4704 y Fp(10)3316 4814 y Fi(\003)3197 4940 y Fp(6)-163 b(6)g(6)g(3)2577 5051 y(\(f)7 b(\))1397 5260 y(Fig.)31 b(4:)41 b(An)m(tipath)30 b(\014nding)1831 5652 y(14)p eop %%Page: 15 15 15 14 bop 827 1472 a 3.3 setlinewidth 2185 1236 4 709 v 2187 529 709 4 v 2894 1236 4 709 v 2187 1238 709 4 v 2187 647 V 2187 765 V 2187 883 V 2187 1002 V 2187 1120 V 2303 1236 4 709 v 2421 1236 V 2539 1236 V 2658 1236 V 2776 1236 V 2058 599 a Fk(x)2110 613 y Fh(5)2058 717 y Fk(x)2110 731 y Fh(6)2058 836 y Fk(x)2110 850 y Fh(7)2058 954 y Fk(x)2110 968 y Fh(8)2058 1072 y Fk(x)2110 1086 y Fh(9)2041 1190 y Fk(x)2093 1204 y Fh(10)2188 469 y Fk(x)2240 483 y Fh(5)2306 469 y Fk(x)2358 483 y Fh(6)2424 469 y Fk(x)2476 483 y Fh(7)2543 469 y Fk(x)2595 483 y Fh(8)2661 469 y Fk(x)2713 483 y Fh(9)2761 469 y Fk(x)2813 483 y Fh(10)2814 734 y Fp(1)-164 b(1)h(1)2600 704 y Fd(j)2814 1088 y Fp(1)2836 1059 y Fd(j)2932 608 y Fi(\003)2932 844 y(\003)2932 962 y(\003)2932 1198 y(\003)1006 1213 y Fd(x)161 b Fc(\000)1325 1130 y(\000)1395 1059 y(\000)1437 823 y(\001)1395 906 y(\001)1354 989 y(\001)1312 1072 y(\001)1271 1155 y(\001)1242 1212 y(\001)1242 976 y Fd(x)1242 1213 y(x)1478 976 y(x)1478 740 y(x)1714 1213 y(x)1031 1296 y Fk(x)1083 1310 y Fh(5)1267 1296 y Fk(x)1319 1310 y Fh(6)1763 1296 y Fk(x)1815 1310 y Fh(7)1102 918 y Fk(x)1154 932 y Fh(8)1550 918 y Fk(x)1602 932 y Fh(9)1462 658 y Fk(x)1514 672 y Fh(10)1255 1448 y Fp(\(a\))1110 b(\(b\))p 1466 976 25 237 v 1229 1213 V 781 1668 a(Fig.)31 b(5:)41 b(The)30 b(directed)g(subgraph)e(induced)h(b)m(y)h(the)g(la)m(y)m(er)h Fk(X)2931 1635 y Fh(1)827 2801 y 3.3 setlinewidth 2185 2565 4 709 v 2187 1858 709 4 v 2894 2565 4 709 v 2187 2567 709 4 v 2187 1976 V 2187 2094 V 2187 2212 V 2187 2331 V 2187 2449 V 2303 2565 4 709 v 2421 2565 V 2539 2565 V 2658 2565 V 2776 2565 V 2058 1928 a Fk(x)2110 1942 y Fh(5)2058 2046 y Fk(x)2110 2060 y Fh(6)2058 2164 y Fk(x)2110 2178 y Fh(7)2058 2283 y Fk(x)2110 2297 y Fh(8)2058 2401 y Fk(x)2110 2415 y Fh(9)2041 2519 y Fk(x)2093 2533 y Fh(10)2188 1798 y Fk(x)2240 1812 y Fh(5)2306 1798 y Fk(x)2358 1812 y Fh(6)2424 1798 y Fk(x)2476 1812 y Fh(7)2543 1798 y Fk(x)2595 1812 y Fh(8)2661 1798 y Fk(x)2713 1812 y Fh(9)2761 1798 y Fk(x)2813 1812 y Fh(10)2814 2063 y Fp(1)2836 2033 y Fd(j)2341 2299 y Fp(1)2364 2269 y Fd(j)2341 2417 y Fp(1)428 b(1)2822 2301 y(f)2932 1937 y Fi(\003)2909 2063 y Fp(10)2932 2173 y Fi(\003)2932 2299 y Fp(6)2932 2409 y Fi(\003)2932 2527 y(\003)2814 2653 y Fp(9)-518 b(9)1375 2777 y(\(a\))969 b(\(b\))1006 2497 y Fd(x)1242 2107 y Fc(S)1304 2190 y(S)1366 2273 y(S)1429 2357 y(S)1491 2440 y(S)1534 2497 y(S)-80 b(Q)1433 2442 y(Q)1350 2386 y(Q)1267 2331 y(Q)1245 2316 y(Q)1242 2024 y Fd(x)1242 2261 y(x)1242 2497 y(x)279 b(x)161 b(x)1031 2580 y Fk(x)1083 2594 y Fh(5)1267 2580 y Fk(x)1319 2594 y Fh(8)1621 2580 y Fk(x)1673 2594 y Fh(9)1858 2580 y Fk(x)1910 2594 y Fh(7)1078 2226 y Fk(x)1130 2240 y Fh(6)1060 1943 y Fk(x)1112 1957 y Fh(10)p 1229 2497 25 473 v 1185 2997 a Fp(Fig.)30 b(6:)41 b(The)30 b(new)g(directed)g(subgraph)382 3272 y(and)1505 3385 y Fi(U)9 b Fp(\()p Fk(x)1658 3399 y Fh(6)1697 3385 y Fp(\))26 b(=)f Fi(f)p 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Fh(0)3310 4568 y Fp(\).)382 4681 y(The)25 b(adjacen)m(t)i(matrix)f(of)1327 4658 y Fk(~)1314 4681 y(G)1386 4695 y Fj(t)1415 4681 y Fp(\()p Fk(X)1532 4648 y Fh(1)1572 4681 y Fp(\))g(is)f(sho)m(wn)g(in)g(Fig.)39 b(5)26 b(\(b\).)40 b(The)25 b(general)h(an)m(tipath)382 4794 y Fi(U)439 4808 y Fh(1)509 4794 y Fp(of)626 4771 y Fk(~)612 4794 y(G)684 4808 y Fj(t)713 4794 y Fp(\()p Fk(X)830 4761 y Fh(1)870 4794 y Fp(\))31 b(equals)1448 4907 y Fi(U)1505 4921 y Fh(1)1570 4907 y Fp(=)25 b Fi(f)p Fk(x)1763 4921 y Fh(5)1803 4907 y Fk(;)15 b(x)1895 4921 y Fh(7)1935 4907 y Fk(;)g(x)2027 4921 y Fh(8)2066 4907 y Fk(;)g(x)2158 4921 y Fh(10)2233 4907 y Fi(g)p Fk(:)523 5074 y Fp(This)23 b(an)m(tipath)h(is)f(a)i(MIS)f(of)1542 5051 y Fk(~)1529 5074 y(G)o Fp(\()p Fk(X)1717 5041 y Fh(1)1757 5074 y Fp(\);)j(ho)m(w)m(ev)m(er,)g Fk(C)7 b(ar)s(d)p Fp(\()p Fi(U)2516 5088 y Fh(1)2556 5074 y Fp(\))25 b Fk(>)g(C)7 b(ar)s(d)p Fp(\()p Fk(X)3040 5041 y Fh(1)3080 5074 y Fp(\).)39 b(Con-)382 5187 y(sequen)m(tly)-8 b(,)26 b(the)g(directed)e(subgraph)1695 5164 y Fk(~)1681 5187 y(G)p Fp(\()p Fk(X)1870 5154 y Fh(1)1910 5187 y Fp(\))h(is)g(not)g(v)m(ertex-saturated)i(with)d(resp)s(ect)382 5300 y(to)31 b(its)f(initiating)e(set.)1831 5652 y(15)p eop %%Page: 16 16 16 15 bop 523 548 a Fp(Therefore,)39 b(w)m(e)e(assume)g Fk(W)49 b Fp(=)36 b Fi(f)p Fk(x)1769 562 y Fh(5)1809 548 y Fk(;)15 b(x)1901 562 y Fh(7)1941 548 y Fk(;)g(x)2033 562 y Fh(8)2073 548 y Fk(;)g(x)2165 562 y Fh(10)2240 548 y Fi(g)37 b Fp(and)g(reorien)m(tate)h(all)e(arcs)h(of)395 638 y Fk(~)382 661 y(G)o Fp(\()p Fk(X)570 628 y Fh(1)611 661 y Fp(\))32 b(incoming)f(to)i(the)f(v)m(ertices)h(of)g Fk(W)13 b Fp(.)45 b(W)-8 b(e)34 b(obtain)d(a)i(new)f(directed)f (subgraph)382 774 y(represen)m(ted)c(in)e(Fig.)39 b(6)28 b(\(a\).)40 b(Clearly)-8 b(,)27 b(this)e(subgraph)g(has)i(a)g(new)f (initiating)f(set)i Fk(X)3331 741 y Fh(1)382 887 y Fp(=)j Fi(f)p Fk(x)580 901 y Fh(5)620 887 y Fk(;)15 b(x)712 901 y Fh(7)752 887 y Fk(;)g(x)844 901 y Fh(8)883 887 y Fk(;)g(x)975 901 y Fh(10)1050 887 y Fi(g)p Fp(.)523 1000 y(Examining)32 b(as)i(ab)s(o)m(v)m(e,)i(w)m(e)f(\014nd)d(that)j (the)f(new)f(directed)g(subgraph)3025 977 y Fk(~)3012 1000 y(G)o Fp(\()p Fk(X)3200 967 y Fh(1)3240 1000 y Fp(\))i(is)382 1112 y(v)m(ertex-saturated)d(with)d(resp)s(ect)i(to)g(its)f(initiating) d(set)k Fk(X)2436 1080 y Fh(1)2476 1112 y Fp(.)523 1225 y(Th)m(us,)g(w)m(e)h(ma)m(y)g(construct)g(a)g(new)f(digraph)2128 1202 y Fk(~)2114 1225 y(G)p Fp(\()p Fk(X)2303 1192 y Fh(0)2343 1225 y Fp(\).)45 b(The)31 b(adjacen)m(t)i(matrix)d(of)382 1338 y(this)c(digraph)f(can)i(b)s(e)f(obtained)g(from)g(the)h(adjacen)m (t)h(matrix)e(of)h(the)g(initial)d(digraph)382 1451 y(if)k(the)i (corresp)s(onding)d(part)j(of)f(it)g(is)g(replaced)f(b)m(y)i(the)f (adjacen)m(t)i(matrix)e(of)3095 1428 y Fk(~)3081 1451 y(G)p Fp(\()p Fk(X)3270 1418 y Fh(1)3310 1451 y Fp(\).)523 1564 y(Similarly)-8 b(,)35 b(w)m(e)i(determine)e(that)i(a)g(directed)f (subgraph)2559 1541 y Fk(~)2546 1564 y(G)o Fp(\()p 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Fp(\))34 b Fg(=)f Fp(\()p Fk(Y)1574 4568 y Fj(k)1617 4553 y Fk(;)15 b Fp(\000\))34 b(\()p Fk(k)d Fp(=)p 1995 4484 166 4 v 28 w(0)p Fk(;)15 b(m)p Fp(\))34 b Fg(b)-5 b(e)34 b(a)g(vertex-satur)-5 b(ate)g(d)36 b(digr)-5 b(aph)382 4666 y(with)31 b(r)-5 b(esp)g(e)g(ct)31 b(to)g(the)f(initiating)g(set)g Fk(X)1738 4633 y Fj(k)1781 4666 y Fg(.)41 b(Then)30 b(e)-5 b(ach)31 b(antip)-5 b(ath)32 b Fi(U)j(\032)25 b Fk(Y)2877 4681 y Fj(k)2933 4666 y Fi(n)14 b Fk(X)3074 4633 y Fj(k)3148 4666 y Fg(of)30 b(the)382 4779 y(gr)-5 b(aph)643 4756 y Fk(~)629 4779 y(G)701 4793 y Fj(t)730 4779 y Fp(\()p Fk(X)847 4746 y Fj(k)891 4779 y Fp(\))33 b Fg(=)f Fp(\()p Fk(Y)1149 4794 y Fj(k)1192 4779 y Fk(;)15 b Fp(\000)1289 4793 y Fj(t)1319 4779 y Fp(\))33 b Fg(ob)-5 b(eys)33 b(the)g(fol)5 b(lowing)33 b(r)-5 b(elation)1280 4974 y Fk(C)7 b(ar)s(d)p Fp(\()p Fk(X)1608 4936 y Fj(k)1671 4974 y Fi(\\)20 b Fp(\000)1809 4935 y Fv(\000)p Fh(1)1809 4997 y Fj(t)1903 4974 y Fi(U)9 b Fp(\))26 b Fi(\025)f Fk(C)7 b(ar)s(d)p Fp(\()p Fi(U)i Fp(\))782 b(\(2\))523 5177 y(Assume)25 b(that)h(an)f(an)m(tipath)g Fi(U)34 b(\032)25 b Fk(Y)1769 5192 y Fj(k)1821 5177 y Fi(n)10 b Fk(X)1958 5144 y Fj(k)2027 5177 y Fp(of)2138 5154 y Fk(~)2125 5177 y(G)2197 5191 y Fj(t)2226 5177 y Fp(\()p Fk(X)2343 5144 y Fj(k)2386 5177 y Fp(\))26 b(=)f(\()p Fk(Y)2631 5192 y Fj(k)2674 5177 y Fk(;)15 b Fp(\000)2771 5191 y Fj(t)2800 5177 y Fp(\))26 b(will)c(b)s(e)j(found)382 5290 y(suc)m(h)30 b(that)1232 5403 y Fk(C)7 b(ar)s(d)p Fp(\()p Fk(X)1560 5366 y Fj(k)1623 5403 y Fi(\\)20 b Fp(\000)1761 5365 y Fv(\000)p Fh(1)1761 5426 y Fj(t)1855 5403 y Fp(\()p Fi(U)9 b Fp(\)\))26 b Fk(<)f(C)7 b(ar)s(d)p Fp(\()p Fi(U)i Fp(\))p Fk(:)1831 5652 y Fp(16)p eop %%Page: 17 17 17 16 bop 523 548 a Fp(Construct)20 b(a)h(set)g Fi(U)1202 515 y Fv(\003)1262 548 y Fp(=)f(\()p Fk(X)1470 515 y Fj(k)1514 548 y Fi(n)q Fp(\000)1617 510 y Fv(\000)p Fh(1)1617 571 y Fj(t)1712 548 y Fp(\()p Fi(U)9 b Fp(\)\))q Fi([)q(U)g Fp(.)37 b(Clearly)-8 b(,)21 b(the)g(set)g Fi(U)2744 515 y Fv(\003)2804 548 y Fp(is)f(indep)s(enden)m(t)382 661 y(in)29 b(the)i(v)m(ertex-saturated)h(sub)s(digraph)1811 638 y Fk(~)1798 661 y(G)o Fp(\()p Fk(X)1986 628 y Fj(k)2030 661 y Fp(\))e(and)g Fk(C)7 b(ar)s(d)p Fp(\()p Fi(U)2584 628 y Fv(\003)2623 661 y Fp(\))26 b Fk(>)f(C)7 b(ar)s(d)p Fp(\()p Fk(X)3108 628 y Fj(k)3151 661 y Fp(\).)523 774 y(Since)29 b(the)h(set)h Fk(X)1140 741 y Fj(k)1213 774 y Fp(is)e(an)h(an)m(tipath)f(of)i(the)f(TCG)2301 751 y Fk(~)2287 774 y(G)2359 788 y Fj(t)2388 774 y Fp(\()p Fk(X)2505 741 y Fj(k)2549 774 y Fp(\))g(=)g(\()p Fk(Y)2803 789 y Fj(k)2846 774 y Fk(;)15 b Fp(\000)2943 788 y Fj(t)2972 774 y Fp(\))31 b(then)e(the)382 887 y(set)i Fk(X)606 854 y Fj(k)669 887 y Fi(n)21 b Fp(\000)792 848 y Fv(\000)p Fh(1)792 909 y Fj(t)886 887 y Fp(\()p Fi(U)9 b Fp(\))26 b Fi(\032)f Fk(X)1226 854 y Fj(k)1299 887 y Fp(is)30 b(an)g(an)m(tipath)g(of)h(this)e(graph.)40 b(The)30 b(set)h Fi(U)j(\032)25 b Fk(Y)3026 902 y Fj(k)3089 887 y Fi(n)20 b Fk(X)3236 854 y Fj(k)3310 887 y Fp(is)382 1000 y(an)30 b(an)m(tipath)g(of)991 977 y Fk(~)978 1000 y(G)1050 1014 y Fj(t)1079 1000 y Fp(\()p Fk(X)1196 967 y Fj(k)1239 1000 y Fp(\))h(b)m(y)f(conditions)f(of)i(Theorem)f(4.)523 1112 y(Ob)m(viously)-8 b(,)27 b Fi(U)d(\\)16 b Fp(\000)1181 1126 y Fj(t)1210 1112 y Fp(\()p Fk(X)1327 1080 y Fj(k)1386 1112 y Fi(n)g Fp(\000)1504 1074 y Fv(\000)p Fh(1)1504 1135 y Fj(t)1598 1112 y Fp(\()p Fi(U)9 b Fp(\)\))26 b(=)f Fi(\013)p Fp(,)j(therefore)h(the)f(set)h Fi(U)2755 1080 y Fv(\003)2822 1112 y Fp(is)e(an)h(an)m(tipath)382 1225 y(of)499 1202 y Fk(~)485 1225 y(G)557 1239 y Fj(t)587 1225 y Fp(\()p Fk(X)704 1192 y Fj(k)747 1225 y Fp(\).)523 1338 y(W)-8 b(e)31 b(ha)m(v)m(e)h(obtained)d(a)i(con)m(tradiction)f (since)f(the)h(directed)g(subgraph)3026 1315 y Fk(~)3012 1338 y(G)p Fp(\()p Fk(X)3201 1305 y Fj(k)3244 1338 y Fp(\))h(is)382 1451 y(v)m(ertex-saturated)h(with)d(resp)s(ect)i(to)g (its)f(initiating)d(set)k(b)m(y)g(Theorem)f(4.)158 b(Q.E.D.)382 1663 y Fq(Theorem)34 b(5)46 b Fg(L)-5 b(et)30 b(a)g(digr)-5 b(aph)1486 1640 y Fk(~)1473 1663 y(G)o Fp(\()p Fk(X)1661 1630 y Fh(0)1701 1663 y Fp(\))30 b Fg(b)-5 b(e)30 b(vertex-satur)-5 b(ate)g(d.)43 b(Then)30 b(ther)-5 b(e)31 b(exists)f(an)382 1776 y(MPP)g Fi(P)39 b Fg(of)32 b(the)g(gr)-5 b(aph)1236 1753 y Fk(~)1222 1776 y(G)1294 1790 y Fj(t)1323 1776 y Fp(\()p Fk(X)1440 1743 y Fh(0)1480 1776 y Fp(\))32 b Fg(such)g(that)g(its)g(p)-5 b(aths)33 b(c)-5 b(ontain)33 b(only)f(essential)h(ar)-5 b(cs.)523 1987 y Fp(The)27 b(v)-5 b(alidit)m(y)26 b(of)i(this)e(assertion)h(follo)m(ws)g(from)g (the)h(de\014nition)d(of)j(a)f(VS-digraph)382 2100 y(and)j(Theorem)g (4.)2080 b(Q.E.D.)382 2311 y Fq(Corollary)35 b(1)46 b Fg(L)-5 b(et)1114 2288 y Fk(~)1100 2311 y(G)o Fp(\()p Fk(X)1288 2278 y Fh(0)1329 2311 y Fp(\))31 b Fg(b)-5 b(e)31 b(a)g(VS-digr)-5 b(aph.)43 b(Then)31 b(e)-5 b(ach)32 b(p)-5 b(ath)32 b(of)g(an)f(MPP)f Fi(P)38 b Fg(of)395 2401 y Fk(~)382 2424 y(G)454 2438 y Fj(t)483 2424 y Fp(\()p Fk(X)600 2391 y Fh(0)640 2424 y Fp(\))33 b Fg(is)g(b)-5 b(e)g(gun)32 b(by)h(some)g(vertex)g Fk(x)f Fg(of)h Fk(X)1947 2391 y Fh(0)1987 2424 y Fg(.)523 2735 y Fp(Due)24 b(to)h(this)e(result) g(w)m(e)h(ma)m(y)h Fg(use)h(the)h(adjac)-5 b(ent)29 b(matrix)f(of)2602 2712 y Fk(~)2589 2735 y(G)o Fp(\()p Fk(X)2777 2702 y Fh(0)2817 2735 y Fp(\))f Fg(as)h(a)f(working)382 2848 y(table)e Fp(for)g(determination)f(MPP)g(of)h(the)h(TCG)2027 2825 y Fk(~)2013 2848 y(G)2085 2862 y Fj(t)2114 2848 y Fp(\()p Fk(X)2231 2815 y Fh(0)2271 2848 y Fp(\))g(of)f(a)g (VS-digraph.)37 b(Th)m(us,)25 b(w)m(e)382 2961 y(will)d(supp)s(ose)h (that)i(paths)g(of)f(MPP)h(of)g(the)g(TCG)f(of)h(a)g(VS-digraph)e(are)i (found)e(using)382 3074 y(the)33 b(adjacen)m(t)h(matrix)d(of)i(this)f (digraph.)45 b Fq(That)37 b(is,)h(w)m(e)f(c)m(ho)s(ose)i(only)f(essen)m (tial)382 3187 y(arcs)d(to)g(construct)h(eac)m(h)f(new)f(MPP!)523 3300 y Fp(Certainly)-8 b(,)31 b(w)m(e)h(use)f(the)h(adjacen)m(t)h (matrix)e(of)h(the)g(transitiv)m(e)f(closure)g(graph)g(to)382 3413 y(searc)m(h)g(for)f(the)h(maxim)m(um)e(an)m(tipaths)h Fk(U)10 b Fp(\()p Fk(x)p Fp(\))31 b(of)f(suc)m(h)g(TCG.)382 3624 y Fq(Theorem)k(6)46 b Fg(L)-5 b(et)24 b(a)h(vertex)g Fk(x)g Fg(b)-5 b(elongs)25 b(to)g(a)g(VS-digr)-5 b(aph)2435 3601 y Fk(~)2421 3624 y(G)o Fp(\()p Fk(X)2609 3591 y Fh(0)2650 3624 y Fp(\))p Fg(,)26 b(and)g(let)e(the)h(TCG)395 3714 y Fk(~)382 3737 y(G)454 3751 y Fj(t)483 3737 y Fp(\()p Fk(X)600 3704 y Fh(0)640 3737 y Fp(\))33 b Fg(has)g(no)g(maximum)h (antip)-5 b(ath)35 b Fi(U)9 b Fp(\()p Fk(x)p Fp(\))p Fg(.)42 b(Then)33 b(ther)-5 b(e)33 b(exists)g(sever)-5 b(al)33 b(di\013er)-5 b(ent)382 3850 y(MPPs)31 b(of)776 3827 y Fk(~)763 3850 y(G)835 3864 y Fj(t)864 3850 y Fp(\()p Fk(X)981 3817 y Fh(0)1021 3850 y Fp(\))i Fg(such)f(that)i(the)f(vertex) f Fk(x)g Fg(is)g(the)h(he)-5 b(ad)34 b(of)f(distinct)g(engage)-5 b(d)33 b(ar)-5 b(cs)382 3963 y(b)g(elonging)33 b(to)g(p)-5 b(aths)35 b(of)e(these)g(MPPs.)523 4174 y Fp(Let)42 b(the)f(conditions) f(of)h(Theorem)g(6)h(b)s(e)f(ob)s(ey)m(ed.)73 b(F)-8 b(urther,)44 b(let)d Fk(U)52 b Fp(b)s(e)40 b(some)382 4287 y(an)m(tipath)31 b(of)868 4264 y Fk(~)854 4287 y(G)926 4301 y Fj(t)955 4287 y Fp(\()p Fk(X)1072 4254 y Fh(0)1113 4287 y Fp(\))g(suc)m(h)h(that)g Fk(x)27 b Fi(2)g Fk(U)10 b Fp(.)45 b(W)-8 b(e)33 b(designate)e(the)h(maxim)m(um)f(an)m(tipath) 382 4400 y(preceding)42 b Fk(U)53 b Fp(as)43 b Fi(U)1109 4367 y Fv(\003)1149 4400 y Fp(.)79 b(Supp)s(ose)41 b(that)j(the)f(v)m (ertex)h Fk(x)f Fp(is)f(the)i(head)f(of)g(the)g(same)382 4513 y(engaged)30 b(arc)f Fk(y)f(>)d(x)k Fp(for)f(eac)m(h)i(path)f(of)g (an)m(y)g(MPP)g(of)g(the)g(TCG)2690 4490 y Fk(~)2676 4513 y(G)2748 4527 y Fj(t)2777 4513 y Fp(\()p Fk(X)2894 4480 y Fh(0)2934 4513 y Fp(\).)41 b(Without)382 4626 y(loss)23 b(of)h(generalit)m(y)-8 b(,)25 b(w)m(e)f(ma)m(y)g(assume)g (that)g Fk(y)k Fi(2)d(U)2116 4593 y Fv(\003)2155 4626 y Fp(.)38 b(Then)23 b(w)m(e)h(ha)m(v)m(e)h(a)f(con)m(tradiction)382 4739 y(since)30 b(a)g(set)h Fi(U)888 4706 y Fv(\003)948 4739 y Fi(n)21 b(f)p Fk(y)s Fi(g)f([)g(f)p Fk(x)p Fi(g)31 b Fp(is)f(the)g(maxim)m(um)g(an)m(tipath)g(of)2582 4716 y Fk(~)2568 4739 y(G)2640 4753 y Fj(t)2669 4739 y Fp(\()p Fk(X)2786 4706 y Fh(0)2827 4739 y Fp(\).)206 b(Q.E.D.)523 4951 y(Consider)26 b(the)i(digraph)1399 4928 y Fk(~)1385 4951 y(G)p Fp(\()p Fk(X)1574 4918 y Fh(0)1614 4951 y Fp(\))g(represen)m(ted)g(in)f(Fig.)40 b(7)28 b(\(a\),)i(where)d Fk(X)3042 4918 y Fh(0)3110 4951 y Fp(=)h Fi(f)p Fk(x)3306 4965 y Fh(1)3345 4951 y Fp(,)382 5064 y Fk(x)434 5078 y Fh(2)473 5064 y Fp(,)37 b Fk(x)587 5078 y Fh(3)627 5064 y Fp(,)g Fk(x)741 5078 y Fh(4)780 5064 y Fi(g)p Fp(.)57 b(Ob)m(viously)-8 b(,)35 b(the)h(transitiv)m(e)f(closure)g (graph)2515 5041 y Fk(~)2502 5064 y(G)2574 5078 y Fj(t)2603 5064 y Fp(\()p Fk(X)2720 5031 y Fh(0)2760 5064 y Fp(\))h(of)2953 5041 y Fk(~)2939 5064 y(G)p Fp(\()p Fk(X)3128 5031 y Fh(0)3168 5064 y Fp(\))g(has)382 5177 y(no)g(an)m(tipath)f Fi(U)9 b Fp(\()p Fk(x)1038 5191 y Fh(6)1078 5177 y Fp(\).)57 b(It)36 b(is)e(also)i(easy)g(to)h(see)f(that)h(there)f(are)g (di\013eren)m(t)f(MPPs)g(of)395 5267 y Fk(~)382 5290 y(G)454 5304 y Fj(t)483 5290 y Fp(\()p Fk(X)600 5257 y Fh(0)640 5290 y Fp(\))41 b(suc)m(h)f(that)h(the)g(v)m(ertex)h Fk(x)1643 5304 y Fh(6)1722 5290 y Fp(is)e(the)h(head)f(of)h(distinct)d (arcs)j(b)s(elonging)e(to)382 5403 y(paths)34 b(of)g(these)g(MPPs.)52 b(In)33 b(this)g(case)i(paths)f Fk(x)2095 5417 y Fh(1)2168 5403 y Fp({)g Fk(x)2299 5417 y Fh(8)2373 5403 y Fp({)g Fk(x)2504 5417 y Fh(6)2577 5403 y Fp({)h Fk(x)2709 5417 y Fh(10)2784 5403 y Fp(,)g Fk(x)2896 5417 y Fh(2)2969 5403 y Fp({)f Fk(x)3100 5417 y Fh(5)3140 5403 y Fp(,)h Fk(x)3252 5417 y Fh(3)3325 5403 y Fp({)1831 5652 y(17)p eop %%Page: 18 18 18 17 bop 382 548 a Fk(x)434 562 y Fh(9)473 548 y Fp(,)33 b(and)e Fk(x)761 562 y Fh(4)832 548 y Fp({)i Fk(x)962 562 y Fh(7)1033 548 y Fp(form)e(one)i(MPP)-8 b(,)32 b(and)f(paths)h Fk(x)2154 562 y Fh(1)2225 548 y Fp({)h Fk(x)2355 562 y Fh(8)2394 548 y Fp(,)f Fk(x)2503 562 y Fh(2)2575 548 y Fp({)g Fk(x)2704 562 y Fh(5)2743 548 y Fp(,)h Fk(x)2853 562 y Fh(3)2924 548 y Fp({)f Fk(x)3053 562 y Fh(9)3125 548 y Fp({)g Fk(x)3254 562 y Fh(6)3325 548 y Fp({)382 661 y Fk(x)434 675 y Fh(10)509 661 y Fp(,)k(and)e Fk(x)803 675 y Fh(4)878 661 y Fp({)h Fk(x)1010 675 y Fh(7)1085 661 y Fp(form)f(another)i(MPP)-8 b(.)35 b(The)g(v)m(ertex)h Fk(x)2426 675 y Fh(6)2500 661 y Fp(is)f(the)g(head)g(of)g(the)g(arc)382 774 y(\()p Fk(x)469 788 y Fh(8)509 774 y Fk(;)15 b(x)601 788 y Fh(6)640 774 y Fp(\))29 b(in)e(the)h(\014rst)g(MPP)-8 b(,)28 b(and)g(it)g(is)f(the)h(head)g(of)g(the)h(arc)f(\()p Fk(x)2614 788 y Fh(9)2654 774 y Fk(;)15 b(x)2746 788 y Fh(6)2786 774 y Fp(\))28 b(in)f(the)i(second)382 887 y(MPP)-8 b(.)382 1172 y Fl(5)135 b(An)44 b(Alternating)i(P)l(ath)382 1375 y Fp(Let)555 1352 y Fk(~)542 1375 y(G)o Fp(\()p Fk(X)730 1342 y Fh(0)770 1375 y Fp(\))28 b(b)s(e)e(a)i(VS-digraph.)38 b(F)-8 b(urther,)28 b(let)f Fi(P)35 b Fp(=)26 b Fi(f)p Fk(S)2320 1389 y Fh(1)2360 1375 y Fk(;)15 b(:)g(:)g(:)i(;)e(S)2618 1389 y Fj(p)2657 1375 y Fi(g)28 b Fp(\()p Fk(p)d Fp(=)g Fk(C)7 b(ar)s(d)p Fp(\()p Fk(X)3260 1342 y Fh(0)3300 1375 y Fp(\)\))382 1498 y(b)s(e)30 b(some)g(MPP)h(of)f(the)h(TCG)1477 1475 y Fk(~)1464 1498 y(G)1536 1512 y Fj(t)1565 1498 y Fp(\()p Fk(X)1682 1465 y Fh(0)1722 1498 y Fp(\).)523 1611 y(It)g(is)f(easy)i(to)f(see)h(that)f(in)f(general)h(case)h(the)f (graph)2414 1588 y Fk(~)2400 1611 y(G)2472 1625 y Fj(t)2501 1611 y Fp(\()p Fk(X)2618 1578 y Fh(0)2659 1611 y Fp(\))g(can)g(ha)m(v)m (e)h(sev)m(eral)382 1723 y(MPPs.)40 b(W)-8 b(e)32 b(will)c(determine)h (ho)m(w)i(v)-5 b(arious)29 b(MPPs)h(can)h(b)s(e)f(constructed.)523 1836 y(An)i(arc)i Fk(e)e Fp(of)1023 1813 y Fk(~)1009 1836 y(G)1081 1850 y Fj(t)1110 1836 y Fp(\()p Fk(X)1227 1803 y Fh(0)1268 1836 y Fp(\))h(is)e(called)h Fg(engage)-5 b(d)34 b Fp(or)e Fg(thick)h Fp(if)f(it)g(b)s(elongs)g(to)h(a)g(path)g Fk(S)3333 1850 y Fj(q)382 1949 y Fp(\()p Fk(q)28 b Fp(=)p 582 1881 132 4 v 25 w(1)p Fk(;)15 b(p)q Fp(\))31 b(of)f Fi(P)7 b Fp(.)41 b(Otherwise,)30 b(the)g(arc)h Fk(e)g Fp(is)e(called)h Fg(fr)-5 b(e)g(e)31 b Fp(or)f Fg(thin)p Fp(.)523 2062 y(A)h(path)f(of)951 2039 y Fk(~)937 2062 y(G)1009 2076 y Fj(t)1038 2062 y Fp(\()p Fk(X)1155 2029 y Fh(0)1196 2062 y Fp(\))1069 2259 y Fk(P)38 b Fp(=)25 b(\()p Fk(x)1348 2273 y Fj(i)1372 2282 y Fe(1)1411 2259 y Fk(;)15 b(e)1493 2273 y Fj(i)1517 2282 y Fe(1)1556 2259 y Fk(;)g(x)1648 2273 y Fj(i)1672 2282 y Fe(2)1711 2259 y Fk(;)g(e)1793 2273 y Fj(i)1817 2282 y Fe(2)1857 2259 y Fk(;)g(:)g(:)g(:)i(;)e(x)2111 2273 y Fj(i)2135 2281 y Fb(m)2198 2259 y Fk(;)g(e)2280 2273 y Fj(i)2304 2281 y Fb(m)2367 2259 y Fk(;)g(x)2459 2273 y Fj(i)2483 2282 y Fb(m)p Fe(+1)2623 2259 y Fp(\))p Fk(;)382 2455 y Fp(ha)m(ving)36 b(no)h(rep)s(eating)f(arcs)g Fk(e)1457 2469 y Fj(i)1481 2477 y Fb(g)1559 2455 y Fp(\()p Fk(g)j Fp(=)p 1783 2386 166 4 v 36 w(1)p Fk(;)15 b(m)p Fp(\),)39 b(is)d(called)g Fi(P)7 b Fg(-alternating)38 b Fp(if)d(an)m(y)i(ad-)382 2568 y(jacen)m(t)d(arcs)f Fk(e)884 2582 y Fj(i)908 2590 y Fb(g)949 2568 y Fp(,)h Fk(e)1050 2582 y Fj(i)1074 2591 y Fb(g)r Fe(+1)1225 2568 y Fp(\()p Fk(g)26 b Fp(+)21 b(1)30 b Fi(\024)g Fk(m)p Fp(\))j(of)g Fk(P)46 b Fp(ha)m(v)m(e)34 b(opp)s(osite)e(directions)g(and)g(one)h(of)382 2681 y(them)d(is)g(engaged)h(and)f(the)g(other)h(is)e(free.)523 2794 y(If)h Fk(i)645 2808 y Fh(1)710 2794 y Fp(=)25 b Fk(i)837 2808 y Fj(m)p Fh(+1)1024 2794 y Fp(then)30 b(the)h(path)f Fk(P)43 b Fp(is)30 b(also)g(called)g Fi(P)7 b Fg(-alternating)34 b(cycle)p Fp(.)523 2907 y(The)48 b(heads)g(of)h(the)f(arcs)h(of)g Fk(P)61 b Fp(are)49 b(called)f(the)g Fg(upp)-5 b(er)51 b(vertic)-5 b(es)48 b Fp(of)h(the)g Fi(P)7 b Fp(-)382 3019 y(alternating)30 b(path,)g(and)g(the)g(tails)f(of)i(the)f(arcs)h (of)f Fk(P)43 b Fp(are)31 b(called)e(the)i Fg(lower)i(vertic)-5 b(es)382 3132 y Fp(of)34 b(the)h Fi(P)7 b Fp(-alternating)35 b(path.)52 b(The)34 b(set)h(of)f(all)g(upp)s(er)e(v)m(ertices)j(of)f Fk(P)48 b Fp(w)m(e)35 b(designate)382 3245 y(as)c Fk(X)576 3212 y Fh(+)635 3245 y Fp(\()p Fk(P)13 b Fp(\),)31 b(and)f(the)h(set)g (of)f(all)f(lo)m(w)m(er)i(v)m(ertices)g(of)g Fk(P)43 b Fp(w)m(e)31 b(designate)f(as)h Fk(X)3036 3212 y Fv(\000)3095 3245 y Fp(\()p Fk(P)13 b Fp(\).)523 3358 y(The)30 b Fi(P)7 b Fp(-alternating)31 b(path)f Fk(P)43 b Fp(is)30 b(called)f Fg(r)-5 b(e)g(gular)32 b Fp(if:)518 3539 y Fi(\017)46 b Fp(eac)m(h)32 b(of)e(sets)h Fk(X)1177 3506 y Fh(+)1237 3539 y Fp(\()p Fk(P)13 b Fp(\))31 b(and)e Fk(X)1667 3506 y Fv(\000)1727 3539 y Fp(\()p Fk(P)13 b Fp(\))31 b(has)f(no)g(rep)s (eating)g(v)m(ertices;)518 3724 y Fi(\017)46 b Fp(an)34 b(engaged)g(arc)g(of)g Fi(P)41 b Fp(is)33 b(inciden)m(t)f(to)i(no)g (\\upp)s(er")e(or)i(\\lo)m(w)m(er")h(v)m(ertex)g(if)d(it)609 3837 y(do)s(es)e(not)h(b)s(elong)e(to)i Fk(P)13 b Fp(;)518 4022 y Fi(\017)46 b Fp(exactly)31 b(t)m(w)m(o)h(arcs)f(of)f Fk(P)43 b Fp(are)31 b(inciden)m(t)e(to)i(eac)m(h)h(\\upp)s(er")d(v)m (ertex.)523 4203 y(Notice)38 b(that)f(the)g(same)h(v)m(ertex)g(of)1821 4180 y Fk(~)1808 4203 y(G)1880 4217 y Fj(t)1909 4203 y Fp(\()p Fk(X)2026 4170 y Fh(0)2066 4203 y Fp(\))f(ma)m(y)g(b)s(e)g (met)g(t)m(wice:)54 b(once)38 b(as)f(an)382 4316 y(\\upp)s(er",)29 b(the)i(second)f(time)g(as)h(a)g(\\lo)m(w)m(er")g(v)m(ertex)h(of)e(the) h Fi(P)7 b Fp(-alternating)31 b(path.)382 4521 y Fq(Theorem)j(7)46 b Fg(If)26 b Fi(P)35 b Fg(is)26 b(some)i(MPP)e(of)1775 4498 y Fk(~)1762 4521 y(G)1834 4535 y Fj(t)1863 4521 y Fp(\()p Fk(X)1980 4488 y Fh(0)2020 4521 y Fp(\))h Fg(then)g(its)g(tr) -5 b(ansition)29 b(to)f(a)f(new)g(MPP)382 4634 y Fi(P)452 4601 y Fv(\003)518 4634 y Fg(of)g(the)g(same)g(TCG)g(is)f(p)-5 b(ossible)28 b(if)e(and)i(only)f(if)f(ther)-5 b(e)28 b(is)e(a)h(r)-5 b(e)g(gular)28 b Fi(P)7 b Fg(-alternating)382 4747 y(p)-5 b(ath)34 b(in)704 4724 y Fk(~)691 4747 y(G)763 4761 y Fj(t)792 4747 y Fp(\()p Fk(X)909 4714 y Fh(0)949 4747 y Fp(\))p Fg(.)523 4951 y Fp(Notice)47 b(that)f(an)m(y)f(MPP)h(of) 1607 4928 y Fk(~)1594 4951 y(G)1666 4965 y Fj(t)1695 4951 y Fp(\()p Fk(X)1812 4918 y Fh(0)1852 4951 y Fp(\))g(can)f(b)s(e)g (constructed)h(as)g(a)g(maxim)m(um)382 5064 y(matc)m(hing)32 b(for)g(a)h(bipartite)e(graph)h(with)f(2)p Fk(n)h Fp(v)m(ertices)h (\(see)g(Section)f(2\).)48 b(Therefore,)382 5177 y(the)36 b(transformation)g(to)h(a)g(new)e(maxim)m(um)h(matc)m(hing)g(of)g(the)h (bipartite)e(graph)h(is)382 5290 y(p)s(ossible)27 b(if)h(there)h(is)f (a)i Fi(P)7 b Fp(-alternating)29 b(path)g(transforming)f(the)h (existing)g(matc)m(hing)382 5403 y(to)i(another)g(maxim)m(um)e(matc)m (hing.)1831 5652 y(18)p eop %%Page: 19 19 19 18 bop 523 548 a Fp(It)35 b(is)e(easy)j(to)f(see)g(that)g(an)m(y)g (alternating)f(path)h(of)f(the)h(bipartite)e(graph)h(is)g(as-)382 661 y(signed)g(to)i(a)g(regular)f Fi(P)7 b Fp(-alternating)36 b(path)f(of)g(the)h(TCG)2485 638 y Fk(~)2472 661 y(G)2544 675 y Fj(t)2573 661 y Fp(\()p Fk(X)2690 628 y Fh(0)2730 661 y Fp(\).)56 b(Therefore,)37 b(to)382 774 y(construct)30 b(a)h(new)f(MPP)g Fi(P)1351 741 y Fv(\003)1421 774 y Fp(all)f(engaged)j(arcs)e(of)h Fi(P)7 b Fp(,)31 b(b)s(elonging)d(to)j Fi(P)7 b Fp(-alternating)382 887 y(path,)30 b(ha)m(v)m(e)i(to)f(b)s(e)f (replaced)g(b)m(y)g(the)h(free)g(arcs,)g(and)f(all)f(free)i(arcs)f(of)h Fi(P)7 b Fp(,)31 b(b)s(elonging)382 1000 y(to)g Fi(P)7 b Fp(-alternating)31 b(path,)f(ha)m(v)m(e)i(to)f(b)s(e)f(replaced)g(b)m (y)g(the)g(engaged)i(arcs.)180 b(Q.E.D.)523 1212 y(The)34 b(transformation)h(of)g(some)g(initial)d(MPP)j(to)h(another)f(MPP)g(b)m (y)g(means)f(of)382 1325 y(regular)29 b Fi(P)7 b Fp(-alternating)31 b(path)g(\(cycle\))g(w)m(e)g(call)e(a)i Fg(p)-5 b(ermitte)g(d)33 b Fp(transformation.)718 3148 y 3.3 setlinewidth 897 1772 a Fc(H)980 1814 y(H)1063 1855 y(H)1146 1897 y(H)1229 1938 y(H)1287 1967 y(H)1287 1814 y(\000)1204 1897 y(\000)1133 1967 y(\000)1050 1814 y(\000)967 1897 y(\000)897 1967 y(\000)348 b(A)1287 1884 y(A)1245 1801 y(A)1204 1718 y(A)1162 1635 y(A)1133 1578 y(A)-125 b(\000)967 1661 y(\000)897 1731 y(\000)1054 1581 y(\000)971 1664 y(\000)901 1734 y(\000)1057 1585 y(\000)974 1668 y(\000)904 1738 y(\000)1047 1574 y(\000)964 1657 y(\000)894 1727 y(\000)1043 1571 y(\000)960 1654 y(\000)890 1724 y(\000)897 1731 y Fd(x)1133 1495 y(x)1133 1731 y(x)162 b(x)897 1967 y(x)f(x)h(x)899 2074 y Fk(x)951 2088 y Fh(1)1135 2074 y Fk(x)1187 2088 y Fh(2)1371 2074 y Fk(x)1423 2088 y Fh(3)804 1649 y Fk(x)856 1663 y Fh(4)1040 1649 y Fk(x)1092 1663 y Fh(5)1277 1649 y Fk(x)1329 1663 y Fh(6)1182 1460 y Fk(x)1234 1474 y Fh(7)1680 1772 y Fc(H)1763 1814 y(H)1846 1855 y(H)1929 1897 y(H)2012 1938 y(H)2069 1967 y(H)2069 1814 y(\000)1986 1897 y(\000)1916 1967 y(\000)1833 1814 y(\000)1750 1897 y(\000)1680 1967 y(\000)1680 1731 y Fd(x)f(x)g(x)1680 1967 y(x)g(x)g(x)1681 2074 y Fk(x)1733 2088 y Fh(1)1918 2074 y Fk(x)1970 2088 y Fh(2)2154 2074 y Fk(x)2206 2088 y Fh(3)1587 1649 y Fk(x)1639 1663 y Fh(4)1823 1649 y Fk(x)1875 1663 y Fh(5)2059 1649 y Fk(x)2111 1663 y Fh(6)p 2482 1967 4 237 v 2484 1772 a Fc(H)2567 1814 y(H)2650 1855 y(H)2733 1897 y(H)2816 1938 y(H)2874 1967 y(H)2488 1769 y(H)2571 1810 y(H)2654 1852 y(H)2737 1893 y(H)2820 1935 y(H)2877 1964 y(H)2491 1765 y(H)2574 1807 y(H)2657 1848 y(H)2740 1890 y(H)2823 1931 y(H)2881 1960 y(H)2481 1776 y(H)2564 1817 y(H)2647 1859 y(H)2730 1900 y(H)2813 1942 y(H)2870 1971 y(H)2477 1779 y(H)2560 1821 y(H)2643 1862 y(H)2726 1904 y(H)2809 1945 y(H)2866 1974 y(H)p 2955 1967 V 2874 1814 a(\000)2791 1897 y(\000)2720 1967 y(\000)2877 1817 y(\000)2794 1900 y(\000)2724 1971 y(\000)2881 1821 y(\000)2798 1904 y(\000)2727 1974 y(\000)2870 1810 y(\000)2787 1893 y(\000)2717 1964 y(\000)2866 1807 y(\000)2783 1890 y(\000)2713 1960 y(\000)p 2719 1967 V 2637 1814 a(\000)2554 1897 y(\000)2484 1967 y(\000)2641 1817 y(\000)2558 1900 y(\000)2488 1971 y(\000)2644 1821 y(\000)2561 1904 y(\000)2491 1974 y(\000)2634 1810 y(\000)2551 1893 y(\000)2481 1964 y(\000)2630 1807 y(\000)2547 1890 y(\000)2477 1960 y(\000)2915 1967 y(A)2874 1884 y(A)2832 1801 y(A)2791 1718 y(A)2749 1635 y(A)2720 1578 y(A)-125 b(\000)2554 1661 y(\000)2484 1731 y(\000)2641 1581 y(\000)2558 1664 y(\000)2488 1734 y(\000)2644 1585 y(\000)2561 1668 y(\000)2491 1738 y(\000)2634 1574 y(\000)2551 1657 y(\000)2481 1727 y(\000)2630 1571 y(\000)2547 1654 y(\000)2477 1724 y(\000)2484 1731 y Fd(x)2720 1495 y(x)2720 1731 y(x)162 b(x)2484 1967 y(x)f(x)h(x)2486 2074 y Fk(x)2538 2088 y Fh(1)2722 2074 y Fk(x)2774 2088 y Fh(2)2958 2074 y Fk(x)3010 2088 y Fh(3)2391 1649 y Fk(x)2443 1663 y Fh(4)2627 1649 y Fk(x)2679 1663 y Fh(5)2864 1649 y Fk(x)2916 1663 y Fh(6)2769 1460 y Fk(x)2821 1474 y Fh(7)1078 2179 y Fp(\(a\))659 b(\(b\))688 b(\(c\))p 1417 1851 189 4 v 1523 1849 a Fc(-)p 2196 1851 237 4 v 744 w(-)897 2676 y(\000)980 2593 y(\000)1050 2523 y(\000)1050 2912 y(@)967 2829 y(@)897 2759 y(@)897 2912 y(\001)939 2829 y(\001)980 2746 y(\001)1022 2663 y(\001)1063 2580 y(\001)1092 2523 y(\001)153 b(\000)1204 2606 y(\000)1133 2676 y(\000)1133 2759 y(@)1216 2842 y(@)1287 2912 y(@)p Fd(x)-312 b(x)h(x)897 2676 y(x)161 b(x)1133 2440 y(x)h(x)922 3019 y Fk(x)974 3033 y Fh(1)1159 3019 y Fk(x)1211 3033 y Fh(2)1395 3019 y Fk(x)1447 3033 y Fh(3)804 2594 y Fk(x)856 2608 y Fh(4)1206 2688 y Fk(x)1258 2702 y Fh(5)1135 2358 y Fk(x)1187 2372 y Fh(6)1395 2358 y Fk(x)1447 2372 y Fh(7)1727 2676 y Fc(\000)1810 2593 y(\000)1880 2523 y(\000)153 b(\000)2033 2606 y(\000)1963 2676 y(\000)1963 2759 y(@)2046 2842 y(@)2116 2912 y(@)p Fd(x)-311 b(x)1727 2676 y(x)161 b(x)1963 2440 y(x)g(x)1985 3019 y Fk(x)2037 3033 y Fh(2)2222 3019 y Fk(x)2274 3033 y Fh(3)1634 2594 y Fk(x)1686 2608 y Fh(4)2036 2688 y Fk(x)2088 2702 y Fh(5)1965 2358 y Fk(x)2017 2372 y Fh(6)2222 2358 y Fk(x)2274 2372 y Fh(7)2547 2676 y Fc(\000)2630 2593 y(\000)2701 2523 y(\000)2551 2679 y(\000)2634 2596 y(\000)2704 2526 y(\000)2557 2683 y(\000)2640 2600 y(\000)2710 2530 y(\000)2544 2672 y(\000)2627 2589 y(\000)2697 2519 y(\000)2540 2669 y(\000)2623 2586 y(\000)2694 2515 y(\000)p 2782 2912 4 473 v 2701 2912 a(@)2618 2829 y(@)2547 2759 y(@)2547 2912 y(\001)2589 2829 y(\001)2630 2746 y(\001)2672 2663 y(\001)2713 2580 y(\001)2742 2523 y(\001)p 3018 2912 V 153 w(\000)2854 2606 y(\000)2784 2676 y(\000)2940 2526 y(\000)2857 2609 y(\000)2787 2679 y(\000)2944 2530 y(\000)2861 2613 y(\000)2791 2683 y(\000)2933 2519 y(\000)2850 2602 y(\000)2780 2672 y(\000)2930 2515 y(\000)2847 2598 y(\000)2777 2669 y(\000)2784 2759 y(@)2867 2842 y(@)2937 2912 y(@)2787 2755 y(@)2870 2838 y(@)2940 2908 y(@)2791 2752 y(@)2874 2835 y(@)2944 2905 y(@)2780 2762 y(@)2863 2845 y(@)2933 2915 y(@)2777 2766 y(@)2860 2849 y(@)2930 2919 y(@)3020 2912 y Fd(x)-311 b(x)f(x)2547 2676 y(x)162 b(x)2784 2440 y(x)f(x)2571 3019 y Fk(x)2623 3033 y Fh(1)2807 3019 y Fk(x)2859 3033 y Fh(2)3043 3019 y Fk(x)3095 3033 y Fh(3)2454 2594 y Fk(x)2506 2608 y Fh(4)2856 2688 y Fk(x)2908 2702 y Fh(5)2785 2358 y Fk(x)2837 2372 y Fh(6)3043 2358 y Fk(x)3095 2372 y Fh(7)1070 3124 y Fp(\(d\))714 b(\(e\))i(\(f)7 b(\))p 1441 2746 189 4 v 1547 2744 a Fc(-)p 2270 2746 V 746 w(-)p 885 2912 25 237 v 1121 2912 25 473 v 1357 2912 V 1951 2912 V 2187 2912 V 2535 2912 25 237 v 885 1967 V 1357 1967 V 1121 1967 V 1667 1967 V 2140 1967 V 1904 1967 V 1287 3344 a Fp(Fig.)31 b(8:)41 b(The)29 b(alternating)h(cycles)523 3552 y(Fig.)59 b(8)37 b(\(a\))g(represen)m(ts)g(an)f(MPP)h Fi(P)44 b Fp(of)36 b(the)h(transitiv)m(e)f(closure)g(graph.)58 b(This)382 3664 y(MPP)34 b(is)f(transformed)g(to)h(another)g(MPP)-8 b(,)35 b(represen)m(ted)f(in)e(Fig.)52 b(8)34 b(\(c\),)i(b)m(y)e(means) 382 3777 y(of)40 b(the)g Fi(P)7 b Fp(-alternating)41 b(cycle)f Fk(x)1523 3791 y Fh(1)1603 3777 y Fp({)g Fk(x)1740 3791 y Fh(4)1820 3777 y Fp({)g Fk(x)1957 3791 y Fh(3)2036 3777 y Fp({)h Fk(x)2174 3791 y Fh(6)2253 3777 y Fp({)g Fk(x)2391 3791 y Fh(2)2470 3777 y Fp({)f Fk(x)2607 3791 y Fh(5)2687 3777 y Fp(\(see)h(Fig.)69 b(8)41 b(\(b\)\).)382 3890 y(Ob)m(viously)-8 b(,)29 b(in)g(this)g(case)i(the)f Fi(P)7 b Fp(-alternating)31 b(cycle)f(has)g(the)h(set)f(of)h(upp)s(er)d (v)m(ertices)382 4003 y Fk(X)464 3970 y Fh(+)523 4003 y Fp(\()p Fk(P)13 b Fp(\))30 b(=)f Fi(f)p Fk(x)891 4017 y Fh(4)931 4003 y Fp(,)g Fk(x)1037 4017 y Fh(5)1076 4003 y Fp(,)h Fk(x)1183 4017 y Fh(6)1222 4003 y Fi(g)p Fp(,)g(and)f(the)g (set)g(of)h(lo)m(w)m(er)f(v)m(ertices)h Fk(X)2542 3970 y Fv(\000)2601 4003 y Fp(\()p Fk(P)13 b Fp(\))30 b(=)e Fi(f)p Fk(x)2968 4017 y Fh(1)3008 4003 y Fp(,)i Fk(x)3115 4017 y Fh(2)3154 4003 y Fp(,)g Fk(x)3261 4017 y Fh(3)3300 4003 y Fi(g)p Fp(.)523 4116 y(Fig.)66 b(8)39 b(\(d\))g(represen)m(ts)g (an)g(MPP)g Fi(P)1876 4083 y Fv(0)1938 4116 y Fp(of)g(another)g (transitiv)m(e)g(closure)f(graph.)382 4229 y(This)f(MPP)j(is)e (transformed)g(to)i(another)g(MPP)-8 b(,)40 b(represen)m(ted)f(in)f (Fig.)67 b(8)40 b(\(f)7 b(\),)42 b(b)m(y)382 4342 y(means)30 b(of)h(the)g Fi(P)991 4309 y Fv(0)1015 4342 y Fp(-alternating)g(path)f Fk(x)1775 4356 y Fh(2)1845 4342 y Fp({)h Fk(x)1973 4356 y Fh(5)2043 4342 y Fp({)g Fk(x)2171 4356 y Fh(3)2241 4342 y Fp({)g Fk(x)2369 4356 y Fh(7)2439 4342 y Fp({)g Fk(x)2567 4356 y Fh(5)2637 4342 y Fp({)g Fk(x)2765 4356 y Fh(6)2835 4342 y Fp({)g Fk(x)2963 4356 y Fh(4)3033 4342 y Fp(\(see)g(Fig.)382 4455 y(8)i(\(e\)\).)49 b(In)32 b(this)f(case)j(the)f(set)g(of)g(upp)s(er)d(v)m(ertices)j(is)f Fk(X)2347 4422 y Fh(+)2407 4455 y Fp(\()p Fk(P)13 b Fp(\))33 b(=)f Fi(f)p Fk(x)2781 4469 y Fh(5)2821 4455 y Fp(,)h Fk(x)2931 4469 y Fh(6)2971 4455 y Fp(,)g Fk(x)3081 4469 y Fh(7)3120 4455 y Fi(g)p Fp(,)h(and)382 4568 y(the)29 b(set)g(of)g(lo)m(w)m(er)h(v)m(ertices)f(is)f Fk(X)1515 4535 y Fv(\000)1575 4568 y Fp(\()p Fk(P)13 b Fp(\))29 b(=)g Fi(f)p Fk(x)1942 4582 y Fh(2)1981 4568 y Fp(,)h Fk(x)2088 4582 y Fh(3)2127 4568 y Fp(,)f Fk(x)2233 4582 y Fh(4)2273 4568 y Fp(,)g Fk(x)2379 4582 y Fh(5)2419 4568 y Fi(g)p Fp(.)40 b(Notice)30 b(that)g(the)f(MPP)382 4681 y Fi(P)452 4648 y Fv(0)506 4681 y Fp(also)i(has)f(other)g Fi(P)1158 4648 y Fv(0)1182 4681 y Fp(-alternating)g(paths.)382 4967 y Fl(6)135 b(The)44 b(MPP-problem)382 5170 y Fp(In)27 b(Section)h(5)g(w)m(e)h(ha)m(v)m(e)g(sho)m(wn)e(that)i(the)f(same)g (transitiv)m(e)f(closure)h(graph)f(can)h(ha)m(v)m(e)382 5283 y(sev)m(eral)j(di\013eren)m(t)e(MPPs.)41 b(Therefore)30 b(the)h(follo)m(wing)e(problem)f(arises.)1831 5652 y(19)p eop %%Page: 20 20 20 19 bop 523 548 a Fp(Let)701 525 y Fk(~)687 548 y(G)p Fp(\()p Fk(X)876 515 y Fh(0)916 548 y Fp(\))32 b(b)s(e)g(a)g (VS-digraph,)f(and)g(let)2021 525 y Fk(~)2007 548 y(G)2079 562 y Fj(t)2108 548 y Fp(\()p Fk(X)2225 515 y Fh(0)2265 548 y Fp(\))i(b)s(e)e(its)g(TCG.)h(Supp)s(ose)e(that)382 661 y(an)c(MPP)h Fi(P)34 b Fp(of)949 638 y Fk(~)935 661 y(G)1007 675 y Fj(t)1036 661 y Fp(\()p Fk(X)1153 628 y Fh(0)1193 661 y Fp(\))27 b(has)g(mark)m(ed)f(v)m(ertices,)i(that)g (is,)e Fi(B)s Fp(\()p Fi(P)7 b Fp(\))26 b Fi(6)p Fp(=)f Fi(\013)p Fp(.)39 b(F)-8 b(or)28 b(the)f(giv)m(en)382 774 y(TCG)629 751 y Fk(~)616 774 y(G)688 788 y Fj(t)717 774 y Fp(\()p Fk(X)834 741 y Fh(0)874 774 y Fp(\),)32 b(it)e(is)h(necessary)g(to)h(\014nd)d(an)i(MPP)h Fi(P)2283 741 y Fv(\003)2354 774 y Fp(suc)m(h)f(that)g Fi(B)s Fp(\()p Fi(P)2925 741 y Fv(\003)2965 774 y Fp(\))c(=)f Fi(\013)31 b Fp(if)f(it)382 887 y(is)f(p)s(ossible.)39 b(This)28 b(problem)h(is)g(called)h(the)h Fg(MPP-pr)-5 b(oblem)p Fp(.)523 1000 y(It)29 b(is)f(clear)h(in)f(the)h(giv)m(en)g(con)m(text)i (that)e(if)f(the)h(e\013ectiv)m(e)i(solution)c(to)j(the)f(MPP-)382 1112 y(problem)41 b(can)i(b)s(e)f(found,)i(w)m(e)f(can)g(e\013ectiv)m (ely)g(\(in)f(p)s(olynomial)e(time\))j(solv)m(e)f(the)382 1225 y(MCPP)30 b(\(see.)42 b(Lemma)30 b(1\).)523 1338 y(According)36 b(to)g(Section)g(5,)i(the)e(MPP-problem)f(can)h(b)s(e)g (solv)m(ed,)h(for)f(example,)382 1451 y(b)m(y)c(means)h(of)g (constructing)f(an)h(appropriate)e(alternating)h(path)h(or)f(cycle.)48 b(T)-8 b(o)33 b(un-)382 1564 y(derstand)g(the)h(nature)g(of)g(the)h (di\016culties)c(that)k(arise,)f(w)m(e)h(consider)e(the)h(follo)m(wing) 382 1677 y(instance.)p 608 2317 1 473 v 609 1928 a Fc(S)671 2011 y(S)733 2094 y(S)796 2177 y(S)858 2260 y(S)901 2317 y(S)p 963 2317 V 880 1900 a(\021)797 1956 y(\021)714 2011 y(\021)631 2066 y(\021)609 2081 y(\021)609 2317 y(\023)671 2234 y(\023)733 2151 y(\023)796 2068 y(\023)858 1985 y(\023)901 1928 y(\023)p 1317 2317 V 1234 1900 a(\021)1151 1956 y(\021)1068 2011 y(\021)985 2066 y(\021)963 2081 y(\021)963 2137 y(Q)1046 2192 y(Q)1129 2247 y(Q)1212 2303 y(Q)1234 2317 y(Q)p 1553 2075 1 237 v 1554 1928 a(S)1616 2011 y(S)1678 2094 y(S)1741 2177 y(S)1803 2260 y(S)1846 2317 y(S)p 1908 2317 1 473 v 1825 1900 a(\021)1742 1956 y(\021)1659 2011 y(\021)1576 2066 y(\021)1554 2081 y(\021)p 2262 2317 V 2179 1900 a(\021)2096 1956 y(\021)2013 2011 y(\021)1930 2066 y(\021)1908 2081 y(\021)1908 2137 y(Q)1991 2192 y(Q)2074 2247 y(Q)2157 2303 y(Q)2179 2317 y(Q)1132 1847 y Fd(\034)p 1132 2185 1 207 v 1132 2319 a(\035)p 996 2319 5 1 v 996 1847 V 465 2089 a Fk(x)517 2103 y Fh(4)823 2094 y Fk(x)875 2108 y Fh(5)563 1763 y Fk(x)615 1777 y Fh(6)917 1763 y Fk(x)969 1777 y Fh(7)1272 1763 y Fk(x)1324 1777 y Fh(8)1484 2174 y Fk(x)1536 2188 y Fh(4)1768 2094 y Fk(x)1820 2108 y Fh(5)1508 1763 y Fk(x)1560 1777 y Fh(6)1862 1763 y Fk(x)1914 1777 y Fh(7)2217 1763 y Fk(x)2269 1777 y Fh(8)609 1845 y Fd(v)296 b(v)h(v)609 2081 y(v)f(v)609 2317 y(v)g(v)h(v)1554 1845 y(v)1554 2081 y(v)1908 1845 y(v)f(v)1908 2081 y(v)1908 2317 y(v)g(v)p 2498 2317 1 473 v 2499 1928 a Fc(S)2561 2011 y(S)2623 2094 y(S)2685 2177 y(S)2748 2260 y(S)2791 2317 y(S)2502 1924 y(S)2564 2007 y(S)2627 2090 y(S)2689 2173 y(S)2751 2257 y(S)2794 2314 y(S)2506 1921 y(S)2568 2004 y(S)2630 2087 y(S)2693 2170 y(S)2755 2253 y(S)2798 2310 y(S)2495 1932 y(S)2557 2015 y(S)2620 2098 y(S)2682 2181 y(S)2744 2264 y(S)2787 2321 y(S)2492 1935 y(S)2554 2018 y(S)2616 2101 y(S)2678 2184 y(S)2741 2267 y(S)2784 2324 y(S)p 2853 2317 V 2770 1900 a(\021)2687 1956 y(\021)2604 2011 y(\021)2521 2066 y(\021)2499 2081 y(\021)2766 1897 y(\021)2683 1952 y(\021)2600 2007 y(\021)2517 2063 y(\021)2495 2078 y(\021)2763 1893 y(\021)2680 1949 y(\021)2597 2004 y(\021)2514 2059 y(\021)2492 2074 y(\021)2773 1904 y(\021)2690 1959 y(\021)2607 2015 y(\021)2524 2070 y(\021)2502 2085 y(\021)2777 1907 y(\021)2694 1963 y(\021)2611 2018 y(\021)2528 2073 y(\021)2506 2088 y(\021)2499 2317 y(\023)2561 2234 y(\023)2623 2151 y(\023)2685 2068 y(\023)2748 1985 y(\023)2791 1928 y(\023)p 3207 2317 V 3124 1900 a(\021)3041 1956 y(\021)2958 2011 y(\021)2875 2066 y(\021)2853 2081 y(\021)3121 1897 y(\021)3038 1952 y(\021)2955 2007 y(\021)2872 2063 y(\021)2849 2078 y(\021)3117 1893 y(\021)3034 1949 y(\021)2951 2004 y(\021)2868 2059 y(\021)2846 2074 y(\021)3128 1904 y(\021)3045 1959 y(\021)2962 2015 y(\021)2879 2070 y(\021)2856 2085 y(\021)3131 1907 y(\021)3048 1963 y(\021)2965 2018 y(\021)2882 2073 y(\021)2860 2088 y(\021)2853 2137 y(Q)2936 2192 y(Q)3019 2247 y(Q)3102 2303 y(Q)3124 2317 y(Q)2856 2133 y(Q)2940 2188 y(Q)3023 2244 y(Q)3106 2299 y(Q)3128 2314 y(Q)2860 2129 y(Q)2943 2185 y(Q)3026 2240 y(Q)3109 2295 y(Q)3131 2310 y(Q)2849 2140 y(Q)2932 2195 y(Q)3015 2251 y(Q)3098 2306 y(Q)3121 2321 y(Q)2846 2144 y(Q)2929 2199 y(Q)3012 2254 y(Q)3095 2310 y(Q)3117 2324 y(Q)2989 1847 y Fd(\034)p 2989 2185 1 207 v 2989 2319 a(\035)p 2853 2319 5 1 v 2853 1847 V 2499 1845 a(v)g(v)g(v)2499 2081 y(v)g(v)2499 2317 y(v)g(v)g(v)2335 2094 y Fk(x)2387 2108 y Fh(4)2713 2094 y Fk(x)2765 2108 y Fh(5)2453 1763 y Fk(x)2505 1777 y Fh(6)2807 1763 y Fk(x)2859 1777 y Fh(7)3162 1763 y Fk(x)3214 1777 y Fh(8)555 2433 y Fk(x)607 2447 y Fh(1)933 2433 y Fk(x)985 2447 y Fh(2)1264 2433 y Fk(x)1316 2447 y Fh(3)1878 2433 y Fk(x)1930 2447 y Fh(2)2209 2433 y Fk(x)2261 2447 y Fh(3)2445 2433 y Fk(x)2497 2447 y Fh(1)2823 2433 y Fk(x)2875 2447 y Fh(2)3154 2433 y Fk(x)3206 2447 y Fh(3)921 2576 y Fp(\(a\))827 b(\(b\))813 b(\(c\))p 598 2317 21 473 v 953 2317 V 1307 2317 V 1898 2317 V 1543 2075 21 237 v 2252 2317 21 473 v 2488 2317 21 237 v 1728 2868 a(Fig.)31 b(9:)523 3098 y(Let)40 b(there)g(b)s(e)f(a)h (digraph)d(represen)m(ted)j(in)e(Fig.)68 b(9)40 b(\(a\))h(\(remem)m(b)s (er)e(that)h(the)382 3211 y(orien)m(tation)34 b(of)g(arcs)g(of)f(a)i 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b(means)h(of)f(the)h(found)e(alternating)h(cycle,)i(w)m (e)f(will)d(obtain)h(a)i(new)f(MPP)h(whose)382 5130 y(paths)d(are)h (sho)m(wn)e(b)m(y)i(thic)m(k)f(lines)f(in)g(the)h(initial)e(digraph)g (\(see)k(\014g.)40 b(9)31 b(\(c\)\).)523 5243 y(Ho)m(w)k(can)f(w)m(e)g (\014nd)e(the)i(required)e(alternating)i(path)f(\(cycle\))i(for)f(an)m (y)g(instance)382 5356 y(of)c(the)h(MPP-problem?)1831 5652 y(20)p eop %%Page: 21 21 21 20 bop 523 548 a Fp(Ob)m(viously)-8 b(,)23 b(the)f(path)h(of)f(the)h (MPP)-8 b(,)23 b(con)m(taining)f(the)h(mark)m(ed)f(v)m(ertex)i Fk(x)3024 562 y Fh(6)3063 548 y Fp(,)h(should)382 661 y(b)s(e)c(c)m(hanged.)38 b(It)21 b(can)h(b)s(e)f(done)g(b)m(y)g(pic)m (king)f(some)i(thic)m(k)f(arc,)j(b)s(elonging)c(to)i(a)f(segmen)m(t)382 774 y(of)37 b(the)g(existing)e(path)h(con)m(taining)h(the)f(indep)s (enden)m(t)f(v)m(ertices)i(of)g(the)g(considered)382 887 y(digraph,)30 b(on)h(the)g(alternating)g(path)f(\(cycle\).)45 b(F)-8 b(urther,)31 b(ob)m(viously)-8 b(,)30 b(it)h(is)f(necessary)382 1000 y(to)f(pic)m(k)f(a)h(thin)e(arc)i(of)g(the)f(TCG)g(so)h(that)g (the)g(indep)s(enden)m(t)d(v)m(ertices)j(of)g(the)f(initial)382 1112 y(digraph)g(will)g(not)j(b)s(elong)e(to)i(the)g(same)g(path)f(of)g (a)h(new)f(MPP)-8 b(.)523 1225 y(Th)m(us,)28 b(a)i(natural)e(w)m(a)m(y) i(of)f(constructing)f(an)h(alternating)f(path)h(\(cycle\))h(consists) 382 1338 y(in)f(a)i(sequen)m(tial)e(c)m(hoice)j(of)e(thic)m(k)g(and)g (thin)f(arcs)i(of)f(the)h(TCG.)523 1451 y(Ho)m(w)e(can)f(w)m(e)h(c)m (ho)s(ose)g(thic)m(k)f(and)f(thin)g(arcs)h(in)m(to)g(an)g(alternating)g (path)g(\(cycle\)?)523 1564 y(New)j(paths)f(of)g(the)h(construction)f (MPP)g(are)h(determined)e(b)m(y)h(a)h(c)m(hoice)g(of)f(eac)m(h)382 1677 y(thic)m(k)e(or)g(thin)e(arc)i(for)g(the)g(alternating)g(path)f (\(cycle\).)41 b(So,)29 b(the)f(c)m(hoice)h(of)f(the)g(thic)m(k)382 1790 y(arc)36 b(determines)e(the)h(\\curren)m(t")h(path)g(of)f(the)h (MPP)f(from)g(whic)m(h)f(the)h(c)m(hosen)h(arc)382 1903 y(will)30 b(b)s(e)i(remo)m(v)m(ed,)i(i.e.)48 b(this)31 b(path)i(of)g(the)f(existing)g(MPP)h(that)g(will)d(b)s(e)i(disrupted.) 382 2016 y(The)c(c)m(hoice)i(of)f(the)g(thin)e(arc,)j(whic)m(h)e(is)f (adjacen)m(t)k(to)e(the)g(thic)m(k)g(arc)g(and)f(c)m(hosen)i(on)382 2129 y(the)h(previous)f(step,)i(determines)e(a)i(new)e(direction)g(of)i (the)f(disrupted)e(path,)i(and)g(it)382 2242 y(also)i(determines)g(a)g (new)g(path)g(of)h(the)f(constructing)g(MPP)-8 b(.)34 b(Th)m(us,)g(one)f(can)h(reac)m(h)382 2355 y(the)g(conclusion:)45 b(the)33 b(c)m(hoice)i(of)e(a)h(thic)m(k)f(arc)h(for)f(the)h (alternating)f(path)g(\(cycle\))i(is)382 2467 y(simple)26 b(\(except)k(the)f(c)m(hoice)g(of)g(suc)m(h)f(an)h(arc)g(on)f(the)h (\014rst)f(step\))h({)g(w)m(e)g(are)g(c)m(ho)s(osing)382 2580 y(a)i(thic)m(k)f(arc)h(that)g(is)e(adjacen)m(t)j(to)f(a)g(thin)e (arc,)i(c)m(hosen)f(in)f(the)i(previous)e(step.)523 2693 y(A)45 b(c)m(hoice)g(of)g(a)g(thin)e(arc)i(is)e(a)i(quite)f(di\013eren) m(t)g(matter.)84 b(Since)43 b(this)h(c)m(hoice)382 2806 y(determines)26 b(some)i(new)f(path)h(of)f(the)h(MPP)-8 b(,)28 b(w)m(e)g(can)g(consider)e(it)h(to)i(b)s(e)e(p)s(ermissible)382 2919 y(if)j(the)i(new)f(path)g(of)g(the)h(MPP)f(do)s(es)g(not)h(ha)m(v) m(e)g(indep)s(enden)m(t)d(v)m(ertices)j(other)g(than)382 3032 y(the)27 b(initial)e(MPP)-8 b(.)27 b(P)m(erhaps,)h(it)f(is)f(the)h (only)g(reasonable)g(criterion)f(for)g(the)i(elemen)m(t-)382 3145 y(b)m(y-elemen)m(t)36 b(construction)f(of)h(the)f(alternating)g (path)g(\(cycle\).)57 b(T)-8 b(o)36 b(create)h(v)-5 b(arious)382 3258 y(\\en)m(vironmen)m(ts")28 b(of)f(the)h(thin)e(arc,)j(considered)d (to)j(b)s(e)d(included)f(in)h(the)i(alternating)382 3371 y(path)38 b(\(cycle\),)43 b(as)c(it)f(has)g(b)s(een)h(doing)e(in)h (algorithms)g(of)g(lo)s(cal)h(extraction)g(\(lo)s(cal)382 3484 y(searc)m(h\),)32 b(is)d(inappropriate)f(since)i(it)g(often)h (leads)e(to)i(an)g(unacceptable)f(searc)m(h.)523 3597 y(Th)m(us,)g(w)m(e)g(come)i(to)f(a)g(conception)f(of)h(the)f(p)s (ermissible)d(mo)s(di\014cation)h(of)j(MPP)-8 b(.)382 3709 y(This)30 b(mo)s(di\014cation)g(will)f(b)s(e)j(p)s(ermissible)c (if)i(eac)m(h)j(c)m(hoice)g(of)f(the)g(thin)f(or)g(thic)m(k)h(arc)382 3822 y(for)39 b(an)h(alternating)f(path)g(\(cycle\))i(results)d(in)g (the)i(app)s(earance)g(of)f(new)g(paths)g(of)382 3935 y(the)32 b(TCG)g(partition)f(suc)m(h)h(that)g(w)m(e)h(ha)m(v)m(e)g(no)f (new)g(pairs)f(of)h(indep)s(enden)m(t)e(v)m(ertices)382 4048 y(b)s(elonging)25 b(to)j(the)f(same)h(path)f(of)g(the)g(MPP)-8 b(.)28 b(Consequen)m(tly)-8 b(,)28 b(an)m(y)f(obtained)g(partial)382 4161 y(solution)f(for)i(eac)m(h)h(step)f(of)g(construction)f(of)h(the)g (solution)f(will)e(b)s(e)i(p)s(ermissible)d(if)j(it)382 4274 y(results)i(in)g(p)s(ermissible)e(mo)s(di\014cation)h(of)j(the)f (existing)g(MPP)-8 b(.)523 4387 y(Fig.)56 b(10)37 b(\(a\))f(sho)m(ws)g (the)f(partial)g(solution)f(\(i.e.,)j(the)f(part)g(of)f(the)h (alternating)382 4500 y(cycle\))k(that)f(is)g(formed)f(b)m(y)h(the)h(v) m(ertices)f Fk(x)1961 4514 y Fh(4)2040 4500 y Fp({)g Fk(x)2176 4514 y Fh(6)2255 4500 y Fp({)g Fk(x)2391 4514 y Fh(2)2470 4500 y Fp({)g Fk(x)2606 4514 y Fh(5)2646 4500 y Fp(,)i(and)e(Fig.)67 b(10)39 b(\(b\))382 4613 y(sho)m(ws)30 b(the)h(TCG)e(partition)g(that)i(is)f(induced)e(b)m(y)i (this)g(partial)f(solution.)39 b(It)31 b(should)382 4726 y(b)s(e)39 b(observ)m(ed)h(that)h(the)f(next)h(c)m(hoice)f(of)h(the)f (thin)e(arc)j(\()p Fk(x)2496 4740 y Fh(3)2536 4726 y Fk(;)15 b(x)2628 4740 y Fh(5)2667 4726 y Fp(\))41 b(cannot)f(b)s(e)g (done)382 4839 y(on)34 b(the)h(basis)f(of)h(earlier)e(found)h(elemen)m (ts)h(of)g(the)f(solution)g(b)s(ecause)g(if)g(this)g(arc)h(is)382 4951 y(c)m(hosen,)g(then)f(the)g(path)f Fk(x)1340 4965 y Fh(3)1413 4951 y Fp({)h Fk(x)1544 4965 y Fh(5)1617 4951 y Fp({)g Fk(x)1748 4965 y Fh(7)1821 4951 y Fp(of)g(the)g(TCG)g (will)d(b)s(e)i(formed.)50 b(This)32 b(path)382 5064 y(con)m(tains)e(t)m(w)m(o)h(indep)s(enden)m(t)d(v)m(ertices)j Fk(x)1805 5078 y Fh(3)1844 5064 y Fk(;)15 b(x)1936 5078 y Fh(7)1976 5064 y Fp(.)41 b(T)-8 b(o)30 b(c)m(ho)s(ose)h(this)e(arc,)i (it)e(is)g(necessary)382 5177 y(to)h(kno)m(w)f Fg(in)j(advanc)-5 b(e)33 b(\(to)f(for)-5 b(ese)g(e\))31 b Fp(that)f(the)f(thic)m(k)h(arc) f(\()p Fk(x)2475 5191 y Fh(3)2515 5177 y Fk(;)15 b(x)2607 5191 y Fh(8)2647 5177 y Fp(\))30 b(will)c(b)s(e)j(c)m(hosen)h(in)382 5290 y(the)g(alternating)g(cycle)h(in)e(the)i(future.)523 5403 y(Th)m(us,)36 b(the)g(ab)s(o)m(v)m(e)g(example)f(of)h(searc)m (hing)f(for)g(an)h(MPP)f(of)h(the)g(TCG,)f(whose)1831 5652 y(21)p eop %%Page: 22 22 22 21 bop 382 548 a Fp(paths)26 b(ha)m(v)m(e)h(no)g(indep)s(enden)m(t)d (v)m(ertices)j(of)f(the)g(initial)e(acyclic)i(digraph,)g(sho)m(ws)g (that)382 661 y(the)35 b(solution)e(to)i(suc)m(h)g(problem)e(cannot)i (b)s(e)f(found)g(without)f(consideration)h(of)h(all)382 774 y(w)m(a)m(ys)c(to)g(construct)g(an)f(alternating)g(path)g (\(cycle\).)523 887 y(In)i(other)h(w)m(ords,)h(w)m(e)f(ha)m(v)m(e)h (determined)e(that)h(without)f(a)h(mo)s(di\014cation)e(of)i(the)382 1000 y(initial)21 b(instance,)k(the)f(problem)e({)i(searc)m(hing)f(for) h(MPP)f(that)h(do)s(es)g(not)g(ha)m(v)m(e)h(mark)m(ed)382 1112 y(v)m(ertices)33 b({)g(cannot)h(b)s(e)e(solv)m(ed)h(e\013ectiv)m (ely)g(b)s(ecause)g(for)f(construction)h(of)g(an)f(alter-)382 1225 y(nating)40 b(path)g(\(cycle\),)k(it)c(is)f(generally)h(necessary) g(to)h(lo)s(ok)f(through)g(all)f(p)s(ossible)382 1338 y(v)-5 b(arian)m(ts,)30 b(the)h(n)m(um)m(b)s(er)e(of)i(whic)m(h)e(is)g (exp)s(onen)m(tial.)523 1451 y(Is)38 b(it)f(p)s(ossible)f(to)i(remo)m (v)m(e)i(the)e(di\016cult)m(y)f(that)h(has)g(arisen)f(and)g(to)i(solv)m (e)f(the)382 1564 y(giv)m(en)30 b(problem?)39 b(Our)29 b(idea)h(of)h(a)f(solution)f(consists)h(of)h(the)f(follo)m(wing.)523 1677 y(Let)42 b(there)f(b)s(e)f(a)i(digraph)1518 1654 y Fk(~)1505 1677 y(G)o Fp(\()p Fk(X)1693 1644 y Fh(0)1734 1677 y Fp(\),)i(TCG)2095 1654 y Fk(~)2081 1677 y(G)2153 1691 y Fj(t)2183 1677 y Fp(\()p Fk(X)2300 1644 y Fh(0)2340 1677 y Fp(\))d(that)h(has)f(MPP)g Fi(P)7 b Fp(.)73 b(Let,)382 1790 y(further,)36 b Fi(B)s Fp(\()p Fi(P)7 b Fp(\))35 b Fi(6)p Fp(=)f Fi(\013)p Fp(.)57 b(Find)34 b(some)i(mark)m(ed)g(v)m (ertex)h Fk(x)2332 1804 y Fj(j)2403 1790 y Fi(2)d(B)s Fp(\()p Fi(P)7 b Fp(\).)58 b(The)35 b(v)m(ertex)i Fk(x)3309 1804 y Fj(j)3345 1790 y Fp(,)382 1903 y(b)m(y)i(the)g(de\014nition,)h (is)e(the)h(head)g(of)g(a)h(\014ctitious)e(arc)h Fk(x)2406 1917 y Fj(i)2474 1903 y Fi(\035)h Fk(x)2657 1917 y Fj(j)2694 1903 y Fp(,)h(whose)e(terminal)382 2016 y(v)m(ertices)c Fk(x)767 2030 y Fj(i)830 2016 y Fp(and)f Fk(x)1063 2030 y Fj(j)1134 2016 y Fp(b)s(elong)f(to)j(the)e(same)h(path)g(of)g(the)f (MPP)h Fi(P)7 b Fp(.)54 b(A)35 b(part)f(of)h(this)382 2129 y(path,)30 b(connecting)h(v)m(ertices)g Fk(x)1455 2143 y Fj(i)1513 2129 y Fp(and)f Fk(x)1742 2143 y Fj(j)1779 2129 y Fp(,)g(w)m(e)h(will)d(designate)i(as)h Fk(l)r Fp(\()p Fk(x)2767 2143 y Fj(i)2795 2129 y Fk(;)15 b(x)2887 2143 y Fj(j)2924 2129 y Fp(\).)523 2242 y(If)21 b(w)m(e)h(assume)f (further)f(for)i(searc)m(hing)f(the)h(MPP)f(that)h(w)m(e)g(are)g(using) e(a)i(pro)s(cedure)382 2355 y(of)34 b(partitioning)e(the)i(ordered)g (set)h(in)m(to)f(paths,)h(ob)m(viously)-8 b(,)34 b(w)m(e)h(should)d (\014nd)g(a)j(w)m(a)m(y)382 2467 y(to)i(c)m(hange)g(this)e(ordered)h (set)g(so)h(that)g(the)f(v)m(ertices)h(of)f(a)h(\014ctitious)e(arc)h Fk(x)3068 2481 y Fj(i)3131 2467 y Fi(\035)f Fk(x)3309 2481 y Fj(j)3345 2467 y Fp(,)382 2580 y(men)m(tioned)e(ab)s(o)m(v)m(e,) j(will)31 b(not)i(b)s(elong)g(to)h(the)g(path)f Fk(l)r Fp(\()p Fk(x)2352 2594 y Fj(i)2380 2580 y Fk(;)15 b(x)2472 2594 y Fj(j)2509 2580 y Fp(\).)51 b(In)32 b(other)i(w)m(ords,)g(w)m(e) 382 2703 y(should)22 b(c)m(hange)j(the)f(initial)e(digraph)1706 2680 y Fk(~)1692 2703 y(G)o Fp(\()p Fk(X)1880 2670 y Fh(0)1921 2703 y Fp(\))i(so)g(that)h(it)f(will)d(not)j(con)m(tain)h (the)f(found)382 2816 y(path)30 b Fk(l)r Fp(\()p Fk(x)710 2830 y Fj(i)738 2816 y Fk(;)15 b(x)830 2830 y Fj(j)867 2816 y Fp(\).)523 2929 y(There)25 b(exist)h(v)-5 b(arious)25 b(w)m(a)m(ys)h(to)h(remo)m(v)m(e)g(the)f(found)e(path)i Fk(l)r Fp(\()p Fk(x)2653 2943 y Fj(i)2681 2929 y Fk(;)15 b(x)2773 2943 y Fj(j)2810 2929 y Fp(\).)40 b(One)25 b(of)h(suc)m(h)382 3052 y(w)m(a)m(y)31 b(is)d(to)j(reorien)m(tate)f(an)g(arc)g(of)g(the)g (initial)d(digraph)2367 3029 y Fk(~)2354 3052 y(G)o Fp(\()p Fk(X)2542 3019 y Fh(0)2582 3052 y Fp(\),)k(incoming)d(in)m(to)h(the)382 3165 y(mark)m(ed)h(v)m(ertex)i Fk(x)1033 3179 y Fj(j)1100 3165 y Fp(and)d(b)s(elonging)g(to)i(the)g(path)f Fk(l)r Fp(\()p Fk(x)2284 3179 y Fj(i)2312 3165 y Fk(;)15 b(x)2404 3179 y Fj(j)2441 3165 y Fp(\).)523 3278 y(W)-8 b(e)25 b(examine)e(ho)m(w)h(this)f(w)m(orks)h(in)e(the)i(example)g(considered) e(ab)s(o)m(v)m(e.)40 b(Ob)m(viously)-8 b(,)382 3390 y(in)33 b(the)i(TCG)g(of)g(the)g(initial)d(digraph,)i(there)h(is)f(the)h(mark)m (ed)g(v)m(ertex)h Fk(x)2945 3404 y Fh(6)2984 3390 y Fp(,)g(whic)m(h)e (is)382 3503 y(the)c(head)h(for)f(the)g(\014ctitious)g(arc)g Fk(x)1629 3517 y Fh(1)1694 3503 y Fi(\035)25 b Fk(x)1862 3517 y Fh(6)1902 3503 y Fp(.)40 b(Th)m(us,)30 b(w)m(e)h(\014nd)d(the)j (path)f Fk(l)r Fp(\()p Fk(x)3023 3517 y Fh(1)3063 3503 y Fk(;)15 b(x)3155 3517 y Fh(6)3195 3503 y Fp(\).)523 3616 y(Fig.)43 b(11)32 b(\(a\))h(sho)m(ws)e(the)g(digraph)f(obtained)g (from)h(the)g(previous)f(one)i(\(the)f(Fig.)382 3729 y(9)26 b(\(a\)\))g(b)m(y)g(reorien)m(tation)f(of)g(the)h(arc)g(\()p Fk(x)1768 3743 y Fh(4)1808 3729 y Fk(;)15 b(x)1900 3743 y Fh(6)1939 3729 y Fp(\),)27 b(coming)e(in)m(to)h(the)f(mark)m(ed)h(v)m (ertex)g Fk(x)3306 3743 y Fh(6)3345 3729 y Fp(.)382 3842 y(Clearly)-8 b(,)29 b(the)g(path)h(found)e(earlier,)h Fk(l)r Fp(\()p Fk(x)1755 3856 y Fh(1)1794 3842 y Fk(;)15 b(x)1886 3856 y Fh(6)1926 3842 y Fp(\),)30 b(do)s(es)f(not)h(exist)f (in)g(the)g(new)g(digraph.)523 3955 y(On)k(the)i(other)f(hand,)h(the)f (same)h(result)e(can)h(b)s(e)g(obtained)f(if)g(w)m(e)i(delete)g(\(tem-) 382 4068 y(p)s(orarily\))c(the)j(leading)f(arc)g(from)h(the)f(initial)e (digraph)2390 4045 y Fk(~)2376 4068 y(G)o Fp(\()p Fk(X)2564 4035 y Fh(0)2605 4068 y Fp(\))j(\(see)g(Fig.)50 b(11)35 b(\(b\)\).)p 1441 4803 1 237 v 1380 4803 a Fc(S)1317 4720 y(S)1255 4637 y(S)1193 4554 y(S)1131 4470 y(S)1088 4413 y(S)p 1087 4566 V 2032 4803 1 473 v 882 w(S)2095 4496 y(S)2157 4579 y(S)2219 4662 y(S)2282 4745 y(S)2325 4803 y(S)2036 4410 y(S)2098 4493 y(S)2161 4576 y(S)2223 4659 y(S)2285 4742 y(S)2328 4799 y(S)2040 4406 y(S)2102 4489 y(S)2164 4572 y(S)2226 4655 y(S)2289 4738 y(S)2332 4795 y(S)2029 4417 y(S)2091 4500 y(S)2153 4583 y(S)2216 4666 y(S)2278 4749 y(S)2321 4806 y(S)2025 4420 y(S)2088 4503 y(S)2150 4586 y(S)2212 4669 y(S)2274 4752 y(S)2317 4810 y(S)p 2386 4803 V 2304 4385 a(\021)2221 4441 y(\021)2138 4496 y(\021)2055 4552 y(\021)2032 4566 y(\021)2032 4803 y(\023)2095 4720 y(\023)2157 4637 y(\023)2219 4554 y(\023)2282 4470 y(\023)2325 4413 y(\023)p 2741 4803 V 2658 4385 a(\021)2575 4441 y(\021)2492 4496 y(\021)2409 4552 y(\021)2387 4566 y(\021)2387 4622 y(Q)2470 4677 y(Q)2553 4732 y(Q)2636 4788 y(Q)2658 4803 y(Q)2519 4332 y Fd(\034)p 2519 4670 1 207 v 2519 4804 a(\035)p 2387 4804 1 1 v 2387 4332 V 1088 4330 a(v)1088 4566 y(v)296 b(v)1442 4803 y(v)2032 4330 y(v)2032 4566 y(v)2032 4803 y(v)2387 4330 y(v)2387 4566 y(v)2387 4803 y(v)2741 4330 y(v)2741 4803 y(v)1042 4260 y Fk(x)1094 4274 y Fh(6)967 4637 y Fk(x)1019 4651 y Fh(4)1469 4527 y Fk(x)1521 4541 y Fh(5)1401 4905 y Fk(x)1453 4919 y Fh(2)1987 4260 y Fk(x)2039 4274 y Fh(6)2341 4260 y Fk(x)2393 4274 y Fh(7)2695 4260 y Fk(x)2747 4274 y Fh(8)1869 4600 y Fk(x)1921 4614 y Fh(4)2241 4585 y Fk(x)2293 4599 y Fh(5)1987 4910 y Fk(x)2039 4924 y Fh(1)2341 4910 y Fk(x)2393 4924 y Fh(2)2695 4910 y Fk(x)2747 4924 y Fh(3)1179 5109 y Fp(\(a\))1048 b(\(b\))p 1432 4803 21 237 v 1077 4566 V 2022 4803 V 2376 4566 V 2731 4803 21 473 v 1705 5353 a(Fig.)31 b(10:)1831 5652 y(22)p eop %%Page: 23 23 23 22 bop 382 548 a Fp(Ob)m(viously)-8 b(,)23 b(in)g(this)f(case,)27 b(in)22 b(the)i(new)f(digraph,)g(obtained)g(from)h(the)g(initial)c (digraph)382 661 y(b)m(y)30 b(deleting)g(the)g(arc)h(\()p Fk(x)1247 675 y Fh(4)1287 661 y Fk(;)15 b(x)1379 675 y Fh(6)1418 661 y Fp(\),)31 b(the)g(path)f Fk(l)r Fp(\()p Fk(x)1994 675 y Fh(1)2034 661 y Fk(;)15 b(x)2126 675 y Fh(6)2166 661 y Fp(\))30 b(do)s(es)g(not)h(exist)f(either.)523 774 y(In)42 b(the)g(next)g(sections)h(of)f(the)g(pap)s(er)f(w)m(e)i (will)d(come)j(bac)m(k)g(to)g(this)e(example)382 887 y(again.)382 1173 y Fl(7)135 b(Co-v)l(ertices)382 1376 y Fp(Let)569 1353 y Fk(~)556 1376 y(G)o Fp(\()p Fk(X)744 1343 y Fh(0)784 1376 y Fp(\))42 b(b)s(e)f(a)g(v)m(ertex-saturated)i (digraph)d(and)g Fi(P)49 b Fp(b)s(e)41 b(an)g(MPP)g(of)3065 1353 y Fk(~)3052 1376 y(G)3124 1390 y Fj(t)3153 1376 y Fp(\()p Fk(X)3270 1343 y Fh(0)3310 1376 y Fp(\).)382 1489 y(Supp)s(ose)31 b(that)i(a)g(graph)g Fk(G)c Fi(2)f Fk(L)1531 1503 y Fj(n)1611 1489 y Fp(has)33 b(some)g(o)s(dd)e(cycle)j Fk(C)2480 1504 y Fh(2)p Fj(k)r Fh(+1)2680 1489 y Fp(\()p Fk(k)f Fi(\025)c Fp(2\))34 b(stretc)m(hed)382 1602 y(on)22 b Fk(k)j Fp(paths)d(of)h Fi(P)7 b Fp(,)24 b(and)e(the)h(TCG)1583 1579 y Fk(~)1569 1602 y(G)1641 1616 y Fj(t)1671 1602 y Fp(\()p Fk(X)1788 1569 y Fh(0)1828 1602 y Fp(\))f(has)g(no)g(MPP)h (suc)m(h)f(that)h(the)f(cycle)h Fk(C)3203 1617 y Fh(2)p Fj(k)r Fh(+1)382 1715 y Fp(is)31 b(densely)g(stretc)m(hed)i(\(see)h (Section)e(3\).)47 b(Suc)m(h)32 b(a)g(cycle)h(will)d(b)s(e)h(called)h Fg(r)-5 b(estricte)g(d)5 b(ly)382 1828 y(str)-5 b(etche)g(d)32 b Fp(on)e(the)h(paths)f(of)1405 1805 y Fk(~)1392 1828 y(G)o Fp(\()p Fk(X)1580 1795 y Fh(0)1620 1828 y Fp(\).)523 1941 y(Ob)m(viously)-8 b(,)25 b(if)e(a)j(VS-digraph)d(has)h(an)h(o)s (dd)f(cycle)h(that)g(is)f(restrictedly)g(stretc)m(hed)382 2054 y(on)35 b(its)g(paths,)h(then,)h(for)e(the)g(giv)m(en)g(digraph,)g (the)h(MPP-problem)e(cannot)i(b)s(e)e(re-)382 2166 y(solv)m(ed)c(as)h (there)f(is)g(alw)m(a)m(ys)h(a)f(mark)m(ed)h(v)m(ertex)g(b)s(elonging)e (to)i(the)g(cycle.)523 2279 y(T)-8 b(o)25 b(remo)m(v)m(e)h(this)d (obstacle,)k(w)m(e)e(construct)g(a)g(new)f(digraph,)g(that)h(is,)g(w)m (e)g(actually)382 2392 y(form)m(ulate)30 b(a)h(new)f(problem.)523 2505 y(Supp)s(ose)48 b(w)m(e)h(ha)m(v)m(e)i(rev)m(ealed)f(an)f(o)s(dd)f (cycle)i Fk(C)2305 2520 y Fh(2)p Fj(k)r Fh(+1)2523 2505 y Fp(\()p Fk(k)60 b Fi(\025)d Fp(2\))50 b(restrictedly)382 2618 y(stretc)m(hed)38 b(on)g(the)g(paths)f(of)1458 2595 y Fk(~)1444 2618 y(G)p Fp(\()p Fk(X)1633 2585 y Fh(0)1673 2618 y Fp(\).)63 b(Clearly)-8 b(,)38 b(in)f(this)f(case)j(there)e(are)h (t)m(w)m(o)h(arcs)382 2731 y(of)33 b(the)g(cycle)g Fk(C)939 2746 y Fh(2)p Fj(k)r Fh(+1)1107 2731 y Fp(,)g(b)s(elonging)e(to)i(the)g (same)g(path)g(of)g(the)f(MPP)h Fi(P)40 b Fp(of)33 b(the)g(TCG)395 2821 y Fk(~)382 2844 y(G)454 2858 y Fj(t)483 2844 y Fp(\()p Fk(X)600 2811 y Fh(0)640 2844 y Fp(\).)39 b(Let)25 b(this)f(b)s(e)g (arcs)h Fk(x)1420 2858 y Fj(i)1473 2844 y Fk(>)g(x)1621 2859 y Fj(k)1688 2844 y Fp(and)f Fk(x)1911 2859 y Fj(k)1979 2844 y Fk(>)h(x)2127 2858 y Fj(j)2163 2844 y Fp(.)39 b(Then)24 b(the)g(v)m(ertex)i Fk(x)2931 2858 y Fj(j)2992 2844 y Fp(is)e(mark)m(ed)382 2957 y(in)29 b(the)i(MPP)f Fi(P)7 b Fp(.)523 3070 y(In)28 b(this)f(case,)k(in)c(the)i(digraph)1631 3047 y Fk(~)1617 3070 y(G)p Fp(\()p Fk(X)1806 3037 y Fh(0)1846 3070 y Fp(\))g(w)m(e)g(in)m(tro)s(duce)f(an)g(additional)e(v) m(ertex)k Fk(x)3322 3037 y Fv(0)3345 3070 y Fp(,)382 3183 y(whic)m(h)25 b(will)g(b)s(e)h(called)g(a)h Fg(c)-5 b(o-vertex)27 b Fp(of)g(the)g(rev)m(ealed)h(cycle)f Fk(C)2522 3198 y Fh(2)p Fj(k)r Fh(+1)2717 3183 y Fp(\()p Fk(k)h Fi(\025)d Fp(2\).)41 b(W)-8 b(e)28 b(join)382 3296 y(the)k(co-v)m (ertex)j Fk(x)986 3263 y Fv(0)1041 3296 y Fp(with)c(\()p Fk(k)25 b Fp(+)c(1\))33 b(v)m(ertices)f(of)h(the)f(cycle)g(suc)m(h)g (that)h(the)f(in)m(tro)s(duced)382 3408 y(co-v)m(ertex)43 b(and)d(other)h(v)m(ertices)g(of)g(the)f(cycle)h Fk(C)2138 3423 y Fh(2)p Fj(k)r Fh(+1)2347 3408 y Fp(form)f(an)g(indep)s(enden)m (t)f(set.)382 3521 y(F)-8 b(or)39 b(determinacy)-8 b(,)40 b(w)m(e)e(supp)s(ose)f(that)h(the)h(v)m(ertex)g Fk(x)2293 3488 y Fv(0)2354 3521 y Fp(is)e(joined)g(b)m(y)h(arcs)g(with)f(all)382 3634 y(v)m(ertices)42 b(of)g(the)g(cycle)h Fk(C)1307 3649 y Fh(2)p Fj(k)r Fh(+1)1516 3634 y Fp(ha)m(ving)f(the)g(minim)m(um) d(rank)i(in)f(eac)m(h)j(path)f Fi(P)49 b Fp(of)395 3724 y Fk(~)382 3747 y(G)454 3761 y Fj(t)483 3747 y Fp(\()p Fk(X)600 3714 y Fh(0)640 3747 y Fp(\))32 b(where)f(the)h(v)m(ertices)g (of)g(the)g(cycle)g(are)g(lo)s(cated,)g(and,)f(is)g(also)h(joined)e (with)p 1152 4739 1 473 v 1153 4267 a Fc(\021)1236 4211 y(\021)1319 4156 y(\021)1402 4101 y(\021)1424 4086 y(\021)1149 4263 y(\021)1232 4208 y(\021)1315 4152 y(\021)1398 4097 y(\021)1421 4082 y(\021)1146 4260 y(\021)1229 4204 y(\021)1312 4149 y(\021)1395 4094 y(\021)1417 4079 y(\021)1156 4270 y(\021)1239 4215 y(\021)1322 4159 y(\021)1405 4104 y(\021)1428 4089 y(\021)1160 4274 y(\021)1243 4218 y(\021)1326 4163 y(\021)1409 4108 y(\021)1431 4093 y(\021)1466 4113 y(\001)1424 4196 y(\001)1383 4279 y(\001)1341 4363 y(\001)1300 4446 y(\001)1258 4529 y(\001)1217 4612 y(\001)1175 4695 y(\001)1153 4739 y(\001)p 1507 4739 1 709 v 1445 4736 a(S)1383 4653 y(S)1320 4570 y(S)1258 4487 y(S)1196 4404 y(S)1153 4347 y(S)1448 4733 y(S)1386 4650 y(S)1334 4580 y(S)1452 4732 y(S)1390 4649 y(S)1337 4579 y(S)1441 4740 y(S)1379 4657 y(S)1326 4587 y(S)1438 4746 y(S)1376 4663 y(S)1323 4593 y(S)p 1861 4739 1 473 v 1779 4322 a(\021)1695 4377 y(\021)1612 4433 y(\021)1529 4488 y(\021)1507 4503 y(\021)1775 4318 y(\021)1692 4374 y(\021)1609 4429 y(\021)1526 4484 y(\021)1504 4499 y(\021)1771 4315 y(\021)1688 4370 y(\021)1605 4426 y(\021)1522 4481 y(\021)1500 4496 y(\021)1782 4326 y(\021)1699 4381 y(\021)1616 4436 y(\021)1533 4492 y(\021)1511 4506 y(\021)1786 4329 y(\021)1703 4384 y(\021)1620 4440 y(\021)1537 4495 y(\021)1514 4510 y(\021)1507 4558 y(Q)1590 4614 y(Q)1673 4669 y(Q)1756 4724 y(Q)1779 4739 y(Q)1511 4555 y(Q)1594 4610 y(Q)1677 4665 y(Q)1760 4721 y(Q)1782 4735 y(Q)1514 4551 y(Q)1597 4606 y(Q)1680 4662 y(Q)1763 4717 y(Q)1786 4732 y(Q)1504 4562 y(Q)1587 4617 y(Q)1670 4672 y(Q)1753 4728 y(Q)1775 4743 y(Q)1500 4565 y(Q)1583 4621 y(Q)1666 4676 y(Q)1749 4731 y(Q)1771 4746 y(Q)1671 4032 y Fd( )p 1671 4590 1 410 v 1671 4741 a(!)p 1507 4741 16 1 v 1507 4032 V 1507 4030 a(v)1153 4267 y(v)1507 4503 y(v)-235 b(v)1862 4267 y(v)1153 4739 y(v)296 b(v)h(v)1488 3949 y Fk(x)1540 3963 y Fh(7)1819 4185 y Fk(x)1871 4199 y Fh(8)989 4279 y Fk(x)1041 4293 y Fh(4)1532 4386 y Fk(x)1584 4400 y Fh(5)1257 4634 y Fk(x)1309 4648 y Fh(6)1107 4858 y Fk(x)1159 4872 y Fh(1)1462 4858 y Fk(x)1514 4872 y Fh(2)1816 4858 y Fk(x)1868 4872 y Fh(3)1465 5001 y Fp(\(a\))p 2097 4736 1 237 v 2098 4347 a Fc(S)2160 4430 y(S)2222 4513 y(S)2285 4596 y(S)2347 4679 y(S)2390 4736 y(S)p 2452 4736 1 473 v 2369 4319 a(\021)2286 4375 y(\021)2203 4430 y(\021)2120 4485 y(\021)2098 4500 y(\021)2098 4736 y(\023)2160 4653 y(\023)2222 4570 y(\023)2285 4487 y(\023)2347 4404 y(\023)2390 4347 y(\023)p 2806 4736 V 2723 4319 a(\021)2640 4375 y(\021)2557 4430 y(\021)2474 4485 y(\021)2452 4500 y(\021)2452 4556 y(Q)2535 4611 y(Q)2618 4666 y(Q)2701 4722 y(Q)2723 4736 y(Q)2621 4266 y Fd(\034)p 2621 4604 1 207 v 2621 4738 a(\035)p 2485 4738 5 1 v 2485 4266 V 1954 4508 a Fk(x)2006 4522 y Fh(4)2312 4513 y Fk(x)2364 4527 y Fh(5)2052 4182 y Fk(x)2104 4196 y Fh(6)2406 4182 y Fk(x)2458 4196 y Fh(7)2761 4182 y Fk(x)2813 4196 y Fh(8)2098 4264 y Fd(v)f(v)g(v)2098 4500 y(v)g(v)2098 4736 y(v)g(v)g(v)2044 4852 y Fk(x)2096 4866 y Fh(1)2422 4852 y Fk(x)2474 4866 y Fh(2)2753 4852 y Fk(x)2805 4866 y Fh(3)2407 4995 y Fp(\(b\))p 1142 4739 21 473 v 2087 4736 21 237 v 2442 4736 21 473 v 2796 4736 V 956 5287 a(Fig.)30 b(11:)42 b(Tw)m(o)30 b(w)m(a)m(ys)i(to)f(remo)m(v)m(e)h(the)e (\014ctitious)f(arc)1831 5652 y(23)p eop %%Page: 24 24 24 23 bop 382 548 a Fp(the)36 b(v)m(ertex)i Fk(x)878 563 y Fj(k)920 548 y Fp(.)59 b(All)35 b(the)h(ab)s(o)m(v)m(e)h(men)m (tioned)f(v)m(ertices)h(will)d(b)s(e)h(b)s(eginnings)f(\(tails\))382 661 y(for)c(the)h(new)e(in)m(tro)s(ducing)g(arcs.)523 774 y(Fig.)40 b(12)30 b(\(a\))g(sho)m(ws)e(a)h(v)m(ertex-saturated)i (digraph,)d(in)f(whic)m(h)h(an)g(o)s(dd)g(cycle)h Fk(C)3306 788 y Fh(5)3345 774 y Fp(,)382 887 y(formed)i(b)m(y)h(v)m(ertices)g Fk(x)1199 901 y Fh(1)1270 887 y Fp({)g Fk(x)1399 901 y Fh(5)1470 887 y Fp({)g Fk(x)1599 901 y Fh(9)1670 887 y Fp({)g Fk(x)1799 901 y Fh(2)1870 887 y Fp({)g Fk(x)1999 901 y Fh(6)2039 887 y Fp(,)g(is)f(rev)m(ealed.)45 b(The)31 b(new)g(arcs)h(starts)382 1000 y(at)f(v)m(ertices)g Fk(x)874 1014 y Fh(1)913 1000 y Fp(,)g Fk(x)1021 1014 y Fh(2)1060 1000 y Fp(,)g Fk(x)1168 1014 y Fh(5)1207 1000 y Fp(,)g(and)f(ends)f(at) i(the)g(v)m(ertex)g Fk(x)2242 967 y Fv(0)2266 1000 y Fp(.)523 1112 y(Fig.)57 b(12)36 b(\(b\))g(sho)m(ws)f(the)h(v)m (ertex-saturated)i(digraph)c(in)g(whic)m(h)g(the)i(co-v)m(ertex)382 1225 y Fk(x)434 1192 y Fv(0)487 1225 y Fp(is)30 b(in)m(tro)s(duced)f (for)h(the)g(rev)m(ealed)h(o)s(dd)e(cycle)i Fk(C)2152 1239 y Fh(5)2192 1225 y Fp(.)523 1338 y(The)37 b(undirected)g(cycle)h Fk(C)1472 1352 y Fh(5)1549 1338 y Fp(and)f(the)h(in)m(tro)s(duced)e (co-v)m(ertex)k(are)f(represen)m(ted)382 1451 y(separately)30 b(in)g(Fig.)40 b(12)31 b(\(c\).)p 1865 2188 4 237 v 1805 2035 a Fc(\023)1743 2118 y(\023)1690 2188 y(\023)1808 2035 y(\023)1746 2118 y(\023)1693 2188 y(\023)1812 2035 y(\023)1750 2118 y(\023)1697 2188 y(\023)1801 2035 y(\023)1739 2118 y(\023)1686 2188 y(\023)1798 2035 y(\023)1735 2118 y(\023)1683 2188 y(\023)311 b(B)2028 2105 y(B)2001 2022 y(B)1973 1939 y(B)1945 1856 y(B)1918 1773 y(B)1890 1690 y(B)1887 1680 y(B)-28 b(J)1942 1763 y(J)1997 1846 y(J)2053 1929 y(J)2108 2012 y(J)2163 2095 y(J)2219 2178 y(J)2225 2188 y(J)-57 b(J)2167 2105 y(J)2113 2023 y(J)2226 2188 y(J)2171 2105 y(J)2116 2023 y(J)2230 2188 y(J)2174 2105 y(J)2120 2023 y(J)2219 2188 y(J)2164 2105 y(J)2109 2023 y(J)2216 2188 y(J)2161 2105 y(J)2106 2023 y(J)p 2279 2188 V 2211 2035 a(\045)2142 2118 y(\045)2084 2188 y(\045)2216 2035 y(\045)2147 2118 y(\045)2088 2188 y(\045)2221 2035 y(\045)2152 2118 y(\045)2093 2188 y(\045)2207 2035 y(\045)2137 2118 y(\045)2079 2188 y(\045)2202 2035 y(\045)2133 2118 y(\045)2074 2188 y(\045)-449 b Fd(u)119 b(u)174 b(u)139 b(u)1871 1952 y(u)206 b(u)103 b(u)1879 1597 y(u)1644 2260 y Fk(x)1696 2274 y Fh(1)1829 2260 y Fk(x)1881 2274 y Fh(2)2046 2260 y Fk(x)2098 2274 y Fh(3)2231 2260 y Fk(x)2283 2274 y Fh(4)1912 1988 y Fk(x)1964 2002 y Fh(6)2105 1889 y Fk(x)2157 1903 y Fh(7)2243 1885 y Fk(x)2295 1899 y Fh(8)1930 1602 y Fk(x)1982 1616 y Fh(10)1914 2371 y Fp(\(a\))1655 2188 y Fc(L)1622 2105 y(L)1589 2022 y(L)1556 1939 y(L)1522 1856 y(L)1500 1798 y(L)-33 b(S)1562 1881 y(S)1624 1964 y(S)1686 2047 y(S)1749 2130 y(S)1794 2191 y(S)1504 1798 y(S)1567 1881 y(S)1629 1964 y(S)1691 2047 y(S)1753 2130 y(S)1799 2191 y(S)1509 1798 y(S)1571 1881 y(S)1634 1964 y(S)1696 2047 y(S)1758 2130 y(S)1804 2191 y(S)1496 1798 y(S)1558 1881 y(S)1621 1964 y(S)1683 2047 y(S)1745 2130 y(S)1791 2191 y(S)1492 1798 y(S)1555 1881 y(S)1617 1964 y(S)1679 2047 y(S)1742 2130 y(S)1787 2191 y(S)1678 1952 y(\024)1728 1869 y(\024)1778 1785 y(\024)1828 1702 y(\024)1838 1685 y(\024)1500 1715 y Fd(u)1678 1952 y(u)1396 1804 y Fk(x)1448 1818 y Fh(5)1628 1867 y Fk(x)1680 1881 y Fh(9)p 1566 3138 V 1506 2985 a Fc(\023)1443 3068 y(\023)1391 3138 y(\023)1509 2985 y(\023)1447 3068 y(\023)1394 3138 y(\023)1513 2985 y(\023)1451 3068 y(\023)1398 3138 y(\023)1502 2985 y(\023)1440 3068 y(\023)1387 3138 y(\023)1499 2985 y(\023)1436 3068 y(\023)1384 3138 y(\023)311 b(B)1729 3055 y(B)1702 2972 y(B)1674 2889 y(B)1646 2806 y(B)1619 2723 y(B)1591 2640 y(B)1588 2631 y(B)1588 2630 y(J)1643 2713 y(J)1698 2796 y(J)1754 2879 y(J)1809 2962 y(J)1864 3045 y(J)1920 3128 y(J)1926 3138 y(J)-57 b(J)1868 3055 y(J)1814 2973 y(J)1927 3138 y(J)1872 3055 y(J)1817 2973 y(J)1931 3138 y(J)1875 3055 y(J)1821 2973 y(J)1920 3138 y(J)1865 3055 y(J)1810 2973 y(J)1917 3138 y(J)1862 3055 y(J)1807 2973 y(J)p 1980 3138 V 1912 2985 a(\045)1843 3068 y(\045)1785 3138 y(\045)1917 2985 y(\045)1848 3068 y(\045)1789 3138 y(\045)1922 2985 y(\045)1852 3068 y(\045)1794 3138 y(\045)1908 2985 y(\045)1838 3068 y(\045)1780 3138 y(\045)1903 2985 y(\045)1834 3068 y(\045)1775 3138 y(\045)-449 b Fd(u)119 b(u)174 b(u)139 b(u)1572 2902 y(u)206 b(u)103 b(u)1580 2547 y(u)1345 3210 y Fk(x)1397 3224 y Fh(1)1530 3210 y Fk(x)1582 3224 y Fh(2)1747 3210 y Fk(x)1799 3224 y Fh(3)1932 3210 y Fk(x)1984 3224 y Fh(4)1606 2938 y Fk(x)1658 2952 y Fh(6)1806 2839 y Fk(x)1858 2853 y Fh(7)1944 2835 y Fk(x)1996 2849 y Fh(8)1636 2552 y Fk(x)1688 2566 y Fh(10)1612 3321 y Fp(\(b\))1356 3138 y Fc(L)1323 3055 y(L)1290 2972 y(L)1257 2889 y(L)1223 2806 y(L)1200 2748 y(L)-33 b(S)1263 2831 y(S)1325 2914 y(S)1387 2997 y(S)1450 3080 y(S)1495 3141 y(S)1205 2748 y(S)1267 2831 y(S)1330 2914 y(S)1392 2997 y(S)1454 3080 y(S)1500 3141 y(S)1210 2748 y(S)1272 2831 y(S)1334 2914 y(S)1397 2997 y(S)1459 3080 y(S)1505 3141 y(S)1197 2748 y(S)1259 2831 y(S)1321 2914 y(S)1384 2997 y(S)1446 3080 y(S)1492 3141 y(S)1193 2748 y(S)1256 2831 y(S)1318 2914 y(S)1380 2997 y(S)1442 3080 y(S)1488 3141 y(S)1379 2902 y(\024)1429 2819 y(\024)1479 2736 y(\024)1528 2653 y(\024)1539 2635 y(\024)1200 2665 y Fd(u)1379 2902 y(u)1097 2754 y Fk(x)1149 2768 y Fh(5)1335 2817 y Fk(x)1387 2831 y Fh(9)2379 3138 y Fc(T)2329 3055 y(T)2279 2972 y(T)2261 2941 y(T)2261 2852 y(\032)2344 2789 y(\032)2427 2727 y(\032)2503 2670 y(\032)2586 2673 y(Z)2669 2735 y(Z)2752 2797 y(Z)2818 2846 y(Z)2860 2930 y(\001)2818 3013 y(\001)2777 3096 y(\001)2760 3130 y(\001)p 2429 3139 331 4 v 2429 3138 a(\024)2479 3055 y(\024)2528 2972 y(\024)2547 2941 y(\024)2597 2935 y(T)2647 3018 y(T)2696 3101 y(T)2726 3149 y(T)p 2595 2857 4 242 v 2597 2616 a Fd(u)2266 2852 y(u)281 b(u)2896 2841 y(u)2434 3138 y(u)g(u)2362 3227 y Fk(x)2414 3241 y Fh(9)2719 3227 y Fk(x)2771 3241 y Fh(2)2147 2817 y Fk(x)2199 2831 y Fh(5)2667 2602 y Fk(x)2719 2616 y Fh(1)2905 2807 y Fk(x)2957 2821 y Fh(6)2654 2892 y Fk(x)2706 2859 y Fv(0)2546 3321 y Fp(\(c\))1500 1718 y Fc(\020)1583 1690 y(\020)1666 1663 y(\020)1749 1635 y(\020)1794 1620 y(\020)1490 1718 y(\020)1573 1690 y(\020)1656 1663 y(\020)1739 1635 y(\020)1785 1620 y(\020)1504 1718 y(\020)1587 1690 y(\020)1670 1663 y(\020)1753 1635 y(\020)1799 1620 y(\020)1509 1718 y(\020)1592 1690 y(\020)1675 1663 y(\020)1758 1635 y(\020)1804 1620 y(\020)1497 1718 y(\020)1580 1690 y(\020)1663 1663 y(\020)1746 1635 y(\020)1792 1620 y(\020)1495 1718 y(\020)1578 1690 y(\020)1661 1663 y(\020)1744 1635 y(\020)1790 1620 y(\020)1514 1718 y(\020)1597 1690 y(\020)1680 1663 y(\020)1763 1635 y(\020)1809 1620 y(\020)1518 1718 y(\020)1601 1690 y(\020)1684 1663 y(\020)1767 1635 y(\020)1813 1620 y(\020)1195 2665 y(\020)1278 2638 y(\020)1361 2610 y(\020)1444 2582 y(\020)1502 2563 y(\020)1186 2665 y(\020)1269 2638 y(\020)1352 2610 y(\020)1435 2582 y(\020)1493 2563 y(\020)1192 2665 y(\020)1275 2638 y(\020)1358 2610 y(\020)1441 2582 y(\020)1498 2563 y(\020)1188 2665 y(\020)1271 2638 y(\020)1354 2610 y(\020)1437 2582 y(\020)1495 2563 y(\020)1199 2665 y(\020)1282 2638 y(\020)1365 2610 y(\020)1448 2582 y(\020)1506 2563 y(\020)1202 2665 y(\020)1285 2638 y(\020)1368 2610 y(\020)1451 2582 y(\020)1509 2563 y(\020)1206 2665 y(\020)1289 2638 y(\020)1372 2610 y(\020)1455 2582 y(\020)1513 2563 y(\020)1317 3130 y(l)1234 3061 y(l)1151 2992 y(l)1068 2922 y(l)985 2853 y(l)964 2836 y(l)964 2817 y(b)1047 2867 y(b)1130 2917 y(b)1213 2967 y(b)1296 3017 y(b)1379 3067 y(b)1462 3116 y(b)1480 3127 y(b)969 2790 y(P)1052 2818 y(P)1135 2845 y(P)1219 2873 y(P)1291 2897 y(P)964 2762 y Fd(u)937 2700 y Fk(x)989 2667 y Fv(0)1130 3546 y Fp(Fig.)31 b(12:)42 b(In)m(tro)s(duction)29 b(of)h(the)h(co-v)m(ertex)523 3754 y(Let)37 b Fk(X)7 b Fp(\()p Fk(C)874 3769 y Fh(2)p Fj(k)r Fh(+1)1043 3754 y Fp(\))37 b(b)s(e)f(the)h(set)h(of)f(the)g(v)m (ertices)g(of)g(the)g(cycle)g Fk(C)2734 3769 y Fh(2)p Fj(k)r Fh(+1)2902 3754 y Fp(,)i(and)d Fk(X)3224 3768 y Fh(1)3300 3754 y Fi(\032)382 3867 y Fk(X)7 b Fp(\()p Fk(C)564 3882 y Fh(2)p Fj(k)r Fh(+1)733 3867 y Fp(\))32 b(b)s(e)g(the)g(set)h(of)f(all)f(v)m(ertices)i(of)f Fk(C)1962 3882 y Fh(2)p Fj(k)r Fh(+1)2162 3867 y Fp(that)h(are)f(adjacen)m(t)i (to)e(the)h(in)m(tro-)382 3980 y(duced)d(co-v)m(ertex.)523 4093 y(T)-8 b(o)32 b(complete)g(the)g(construction)g(of)g(the)g(new)f (digraph,)f(w)m(e)j(join)d(the)i(co-v)m(ertex)382 4205 y Fk(x)434 4172 y Fv(0)489 4205 y Fp(with)e(all)g(v)m(ertices)j(of)e (the)h(digraph)1765 4182 y Fk(~)1752 4205 y(G)o Fp(\()p Fk(X)1940 4172 y Fh(0)1980 4205 y Fp(\))g(joined)f(with)f(at)i(least)g (one)g(v)m(ertex)g(of)382 4318 y(the)c(set)h Fk(X)751 4332 y Fh(1)791 4318 y Fp(.)40 b(A)m(t)29 b(that,)h(if)d Fk(x)1339 4332 y Fj(a)1406 4318 y Fi(2)e Fk(X)1567 4332 y Fh(1)1635 4318 y Fp(and)j Fk(x)1862 4333 y Fj(b)1922 4318 y Fi(2)c Fk(X)g Fi(n)17 b Fk(X)2243 4332 y Fh(1)2311 4318 y Fp(and)28 b(there)g(is)g(an)g(arc)h(\()p Fk(x)3167 4332 y Fj(a)3209 4318 y Fk(;)15 b(x)3301 4333 y Fj(b)3335 4318 y Fp(\))382 4431 y(then)34 b(the)g(arc)h(\()p Fk(x)996 4398 y Fv(0)1020 4431 y Fk(;)15 b(x)1112 4446 y Fj(b)1146 4431 y Fp(\))35 b(is)e(in)m(tro)s(duced,)h(and)g(if)f(there)i(is)e(an)h (arc)h(\()p Fk(x)2769 4446 y Fj(b)2804 4431 y Fk(;)15 b(x)2896 4445 y Fj(a)2938 4431 y Fp(\),)35 b(then)f(the)382 4544 y(arc)d(\()p Fk(x)621 4559 y Fj(b)655 4544 y Fk(;)15 b(x)747 4511 y Fv(0)771 4544 y Fp(\))31 b(is)e(in)m(tro)s(duced.)523 4657 y(As)23 b(the)g(result)f(of)i(the)f(made)g(constructions)g(w)m(e)g (will)e(ha)m(v)m(e)j(an)f(extended)g(digraph)395 4747 y Fk(~)382 4770 y(G)454 4737 y Fv(0)477 4770 y Fp(\()p Fk(X)594 4737 y Fh(0)634 4770 y Fp(\))31 b(=)f(\()p Fk(X)918 4737 y Fv(0)942 4770 y Fk(;)981 4747 y(~)982 4770 y Fp(\000)1039 4737 y Fv(0)1062 4770 y Fp(\),)h(where)f Fk(X)1498 4737 y Fv(0)1552 4770 y Fp(=)g Fk(X)e Fi([)19 b(f)p Fk(x)1933 4737 y Fv(0)1957 4770 y Fi(g)p Fp(.)523 4883 y(Clearly)-8 b(,)46 b(in)c(the)i(extended)f(digraph)1922 4860 y Fk(~)1908 4883 y(G)1980 4850 y Fv(0)2003 4883 y Fp(\()p Fk(X)2120 4850 y Fh(0)2160 4883 y Fp(\),)k(the)d(undirected)e(cycle)i Fk(C)3203 4898 y Fh(2)p Fj(k)r Fh(+1)382 4996 y Fp(\()p Fk(k)29 b Fi(\025)c Fp(2\))g(of)804 4973 y Fk(~)791 4996 y(G)o Fp(\()p Fk(X)979 4963 y Fh(0)1020 4996 y Fp(\),)h(con)m(taining)d (the)i(arc)f(\()p Fk(x)1924 5010 y Fj(i)1953 4996 y Fk(;)15 b(x)2045 5010 y Fj(j)2081 4996 y Fp(\),)26 b(is)e(inserted)f(in)m(to)h (the)g(cycle)h Fk(C)3203 5011 y Fh(2)p Fj(k)r Fh(+2)382 5109 y Fp(induced)j(b)m(y)j(the)f(v)m(ertices)h(of)g(the)f(cycle)h Fk(C)1882 5124 y Fh(2)p Fj(k)r Fh(+1)2080 5109 y Fp(and)f(the)h(in)m (tro)s(duced)e(co-v)m(ertex.)523 5222 y(Let)i Fq(F)f Fp(b)s(e)g(some)h Fg(minimum)i(clique)f(p)-5 b(artition)34 b(\(MCP\))c Fp(of)h Fk(G)25 b Fi(2)f Fk(L)2836 5236 y Fj(n)2883 5222 y Fp(.)1831 5652 y(24)p eop %%Page: 25 25 25 24 bop 382 548 a Fq(Theorem)34 b(8)46 b Fg(L)-5 b(et)40 b(ther)-5 b(e)42 b(b)-5 b(e)40 b Fk(G)f Fp(=)g(\()p Fk(X)r(;)15 b Fp(\000\))41 b Fi(2)e Fk(L)2110 562 y Fj(n)2157 548 y Fg(.)65 b(F)-7 b(urther,)43 b(let)e(the)g(gr)-5 b(aph)42 b Fk(G)3238 515 y Fv(0)3300 548 y Fp(=)382 661 y(\()p Fk(X)499 628 y Fv(0)523 661 y Fk(;)15 b Fp(\000)620 628 y Fv(0)644 661 y Fp(\))25 b Fg(b)-5 b(e)25 b(obtaine)-5 b(d)27 b(fr)-5 b(om)26 b(the)g(gr)-5 b(aph)27 b Fk(G)e Fg(by)g(the)g(intr)-5 b(o)g(duction)28 b(of)d(some)h(c)-5 b(o-vertic)g(es.)382 774 y(Then)25 b(an)g(MCP)f Fq(F)1032 741 y Fv(0)1080 774 y Fg(of)h Fk(G)1251 741 y Fv(0)1298 774 y Fg(with)h Fk(C)7 b(ar)s(d)p Fp(\()p Fq(F)p Fp(\))25 b(=)g Fk(C)7 b(ar)s(d)p Fp(\()p Fq(F)2268 741 y Fv(0)2291 774 y Fp(\))25 b Fg(c)-5 b(an)25 b(b)-5 b(e)24 b(found)i(for)f(any)g (MCP)382 887 y Fq(F)32 b Fg(of)h(the)g(gr)-5 b(aph)35 b Fk(G)p Fg(.)523 1097 y Fp(Let)26 b(the)g(graph)e Fk(G)h Fp(=)g(\()p Fk(X)r(;)15 b Fp(\000\))27 b Fi(2)e Fk(L)1697 1111 y Fj(n)1769 1097 y Fp(and)g(the)g(MCP)g Fq(F)h Fp(of)f(the)h (graph)f Fk(G)g Fp(b)s(e)f(giv)m(en.)382 1210 y(F)-8 b(urther,)35 b(let)g(the)f(graph)g Fk(G)1372 1177 y Fv(0)1427 1210 y Fp(=)e(\()p Fk(X)1647 1177 y Fv(0)1671 1210 y Fk(;)15 b Fp(\000)1768 1177 y Fv(0)1791 1210 y Fp(\))35 b(b)s(e)f(obtained)g(from)g(the)g(graph)g Fk(G)g Fp(b)m(y)g(the)382 1323 y(in)m(tro)s(duction)28 b(of)j(some)g(co-v)m(ertices.)523 1435 y(Clearly)-8 b(,)27 b(the)h(relation)f Fk(C)7 b(ar)s(d)p Fp(\()p Fq(F)1651 1402 y Fv(0)1674 1435 y Fp(\))26 b Fi(\025)f Fk(C)7 b(ar)s(d)p Fp(\()p Fq(F)p Fp(\))27 b(is)g(true)g(for)h (an)m(y)g(MCP)f Fq(F)3093 1402 y Fv(0)3144 1435 y Fp(of)g(the)382 1548 y(graph)j Fk(G)712 1515 y Fv(0)734 1548 y Fp(.)523 1661 y(W)-8 b(e)28 b(sho)m(w)f(that)g(an)m(y)h(MCP)e(of)h(the)g(graph)g Fk(G)2077 1628 y Fv(0)2126 1661 y Fp(can)h(b)s(e)e(obtained)g(from)h (an)m(y)g(MCP)382 1774 y Fq(F)j Fp(of)h(the)f(graph)g Fk(G)g Fp(b)m(y)g(joining)e(of)j(cliques)e(of)h Fq(F)h Fp(with)e(the)h(existing)g(co-v)m(ertices.)523 1887 y(Indeed,)c(the)f (co-v)m(ertex)i Fk(x)1429 1854 y Fv(0)1478 1887 y Fp(is)d(joined)g(b)m (y)h(the)g(edges)h(with)e(\()p Fk(k)13 b Fp(+)d(1\))25 b(v)m(ertices)h(of)f(the)382 2000 y(cycle)j Fk(C)669 2015 y Fh(2)p Fj(k)r Fh(+1)865 2000 y Fp(so)g(that)h(it)f(and)f(other)h Fk(k)j Fp(v)m(ertices)e(of)f(the)g(cycle)g(form)g(an)g(indep)s(enden)m (t)382 2113 y(set.)41 b(Hence,)31 b(there)f(is)f(at)i(least)f(one)h(v)m (ertex)g Fk(x)f Fp(of)g(the)g(cycle)g Fk(C)2572 2128 y Fh(2)p Fj(k)r Fh(+1)2770 2113 y Fp(adjacen)m(t)i(to)e(the)382 2226 y(co-v)m(ertex)j Fk(x)826 2193 y Fv(0)879 2226 y Fp(in)c(the)i(MCP)f Fq(F)1449 2193 y Fv(0)1472 2226 y Fp(.)523 2339 y(Let)c Fk(x)f Fp(b)s(e)g(a)h(v)m(ertex)h(of)e(the)h (cycle)g Fk(C)1754 2354 y Fh(2)p Fj(k)r Fh(+1)1922 2339 y Fp(,)g(b)s(elonging)e(to)i(a)g(clique)e Fk(Q)i Fp(of)f(the)h(MCP)382 2452 y Fq(F)448 2419 y Fv(0)471 2452 y Fp(.)52 b(Join)33 b(the)h(co-v)m(ertex)i Fk(x)1357 2419 y Fv(0)1381 2452 y Fp(,)f(adjacen)m(t)g(to)g(the)f(v)m(ertex)i Fk(x)p Fp(,)f(with)e(the)h(clique)f Fk(Q)p Fp(.)51 b(By)382 2565 y(the)26 b(de\014nition)d(of)j(a)g(co-v)m(ertex,)j(the)d(clique)e Fk(Q)h Fp(and)g(the)h(co-v)m(ertex)i Fk(x)2783 2532 y Fv(0)2832 2565 y Fp(together)f(form)382 2677 y(a)k(new)f(clique)f(of)h (the)h(extended)f Fk(G)1623 2645 y Fv(0)1646 2677 y Fp(.)523 2790 y(P)m(erforming)25 b(suc)m(h)g(joining)e(for)i(all)g(co-v)m (ertices)i(of)f(the)f(graph)g Fk(G)2756 2757 y Fv(0)2804 2790 y Fp(with)f(cliques)g(of)382 2903 y(MCP)f Fq(F)p Fp(,)i(w)m(e)f(obtain)e(MCP)i Fq(F)1435 2870 y Fv(0)1481 2903 y Fp(of)f(the)h(graph)f Fk(G)2050 2870 y Fv(0)2096 2903 y Fp(suc)m(h)g(that)h Fk(C)7 b(ar)s(d)p Fp(\()p Fq(F)2796 2870 y Fv(0)2819 2903 y Fp(\))25 b(=)g Fk(C)7 b(ar)s(d)p Fp(\()p Fq(F)3287 2870 y Fv(0)3310 2903 y Fp(\).)382 3016 y(Q.E.D.)382 3226 y Fq(Theorem)34 b(9)46 b Fg(The)32 b(total)j(numb)-5 b(er)33 b(of)f(c)-5 b(o-vertic)g(es)33 b(of)g(the)g(TCG)2699 3203 y Fk(~)2685 3226 y(G)2757 3240 y Fj(t)2786 3226 y Fp(\()p Fk(X)2903 3193 y Fh(0)2944 3226 y Fp(\))f Fg(is)h Fk(O)s Fp(\()p Fk(n)p Fp(\))p Fg(.)523 3436 y Fp(This)i(result)g(follo)m(ws)h(from)g(Theorem)h(8,)h (since)e(the)h(n)m(um)m(b)s(er)e(of)i(cliques)e(in)h(an)382 3549 y(MCP)30 b Fq(F)g Fp(of)h(the)f(graph)g Fk(G)25 b Fi(2)g Fk(L)1482 3563 y Fj(n)1559 3549 y Fp(do)s(es)30 b(not)g(exceed)i Fk(n)p Fp(.)767 b(Q.E.D.)382 3835 y Fl(8)135 b(A)44 b(Correct)i(MPP)382 4038 y Fp(Generally)31 b(the)h(reorien)m(tation)f(of)h(some)g(arc)h(of)e(the)h(digraph)2592 4015 y Fk(~)2578 4038 y(G)p Fp(\()p Fk(X)2767 4005 y Fh(0)2807 4038 y Fp(\),)h(incoming)d(to)382 4151 y(the)43 b(mark)m(ed)f(v)m(ertex,)47 b(ma)m(y)c(giv)m(e)g(a)g(digraph,)h(ha)m (ving)e(a)g(cycle,)k(and,)g(hence,)f(not)382 4264 y(b)s(elonging)30 b(to)i(the)g(set)h Fk(D)s Fp(\()p Fk(G)p Fp(\).)45 b(W)-8 b(e)32 b(will)e(determine)h(the)g(conditions)g(that)h(allo)m(w)f(us)382 4377 y(to)g(a)m(v)m(oid)g(suc)m(h)f(an)g(undesirable)e(result.)523 4490 y(Let)j Fi(P)37 b Fp(b)s(e)30 b(some)g(MPP)g(of)1491 4467 y Fk(~)1477 4490 y(G)1549 4504 y Fj(t)1578 4490 y Fp(\()p Fk(X)1695 4457 y Fh(0)1736 4490 y Fp(\))g(suc)m(h)g(that)g Fi(B)s Fp(\()p Fi(P)7 b Fp(\))27 b Fi(6)p Fp(=)e Fi(\013)p Fp(,)30 b(that)g(is,)g(there)g(is)f(at)382 4603 y(least)i(one)f(mark)m (ed)h(v)m(ertex)g(in)e Fi(P)7 b Fp(.)523 4716 y(Consider)22 b(the)h(mark)m(ed)g(v)m(ertex)i Fk(x)1682 4730 y Fj(j)1744 4716 y Fi(2)f(B)s Fp(\()p Fi(P)7 b Fp(\).)40 b(Supp)s(ose)21 b(that)j(this)e(v)m(ertex)j(b)s(elongs)382 4839 y(to)c(the)g(path)f Fk(S)888 4853 y Fj(q)951 4839 y Fi(2)25 b(P)j Fp(\()p Fk(q)h Fp(=)p 1329 4770 132 4 v 25 w(1)p Fk(;)15 b(p)p Fp(\).)38 b(The)20 b(essen)m(tial)g(arc)h Fk(x)2281 4854 y Fj(k)2349 4839 y Fk(>)k(x)2497 4853 y Fj(j)2554 4839 y Fp(of)2662 4816 y Fk(~)2648 4839 y(G)2720 4853 y Fj(t)2749 4839 y Fp(\()p Fk(X)2866 4806 y Fh(0)2906 4839 y Fp(\),)e(b)s(elonging) 382 4951 y(to)38 b(the)g(path)g Fk(S)940 4965 y Fj(q)977 4951 y Fp(,)i(is)d(called)g Fg(le)-5 b(ading)38 b Fp(for)g(the)g(v)m (ertex)g Fk(x)2362 4965 y Fj(j)2436 4951 y Fi(2)f(B)s Fp(\()p Fi(P)7 b Fp(\).)64 b(If)37 b(the)h(leading)382 5064 y(arc)h(determines)f(the)i(orien)m(ted)e(path)h(of)g(the)h(maxim)m (um)d(length,)k(connecting)e(the)382 5177 y(v)m(ertices)34 b Fk(x)766 5192 y Fj(k)842 5177 y Fp(and)e Fk(x)1073 5191 y Fj(j)1143 5177 y Fp(in)g(digraph)1602 5154 y Fk(~)1589 5177 y(G)o Fp(\()p Fk(X)1777 5144 y Fh(0)1817 5177 y Fp(\))i(\(that)g(is,)f(there)h(do)s(es)f(not)g(exist)g(another)382 5290 y(path)23 b(connecting)g(these)g(v)m(ertices)h(in)e(this)f (digraph\),)j(then)e(the)i(corresp)s(onding)d(MPP)382 5403 y Fi(P)38 b Fp(of)600 5380 y Fk(~)586 5403 y(G)658 5417 y Fj(t)688 5403 y Fp(\()p Fk(X)805 5370 y Fh(0)845 5403 y Fp(\))30 b(is)g(called)f Fg(c)-5 b(orr)g(e)g(ct)33 b Fp(with)c(resp)s(ect)h(to)h(the)g(mark)m(ed)f(v)m(ertex)i Fk(x)2992 5417 y Fj(j)3028 5403 y Fp(.)1831 5652 y(25)p eop %%Page: 26 26 26 25 bop 382 548 a Fq(Theorem)34 b(10)46 b Fg(F)-7 b(or)45 b(any)g(marke)-5 b(d)46 b(vertex)e Fk(x)2004 562 y Fj(j)2087 548 y Fi(2)i(B)s Fp(\()p Fi(P)7 b Fp(\))p Fg(,)47 b(ther)-5 b(e)46 b(always)g(exists)e(an)382 671 y(MPP)31 b(of)i(the)g(gr)-5 b(aph)1138 648 y Fk(~)1125 671 y(G)1197 685 y Fj(t)1226 671 y Fp(\()p Fk(X)1343 638 y Fh(0)1383 671 y Fp(\))33 b Fg(that)h(is)e(c)-5 b(orr)g(e)g(ct)35 b(with)e(r)-5 b(esp)g(e)g(ct)34 b(to)f Fk(x)2686 685 y Fj(j)2723 671 y Fg(.)523 883 y Fp(Indeed,)48 b(let)d Fi(P)53 b Fp(=)45 b Fi(f)p Fk(S)1341 897 y Fh(1)1380 883 y Fk(;)15 b(:)g(:)g(:)i(;)e(S) 1638 897 y Fj(p)1678 883 y Fi(g)p Fp(,)49 b(where)c Fk(p)k Fp(=)h Fk(C)7 b(ar)s(d)p Fp(\()p Fk(X)2619 850 y Fh(0)2659 883 y Fp(\),)49 b(is)44 b(the)h(MPP)g(of)395 983 y Fk(~)382 1006 y(G)454 1020 y Fj(t)483 1006 y Fp(\()p Fk(X)600 973 y Fh(0)640 1006 y Fp(\).)79 b(Supp)s(ose)41 b(that)j(this)e(MPP)h Fi(P)50 b Fp(is)42 b(not)i(correct)g(with)e(resp)s(ect)g(to)i(some)382 1119 y(mark)m(ed)30 b(v)m(ertex)i Fk(x)1033 1133 y Fj(j)1095 1119 y Fi(2)24 b(B)s Fp(\()p Fi(P)7 b Fp(\).)523 1232 y(Let)32 b Fk(x)739 1246 y Fj(j)802 1232 y Fi(2)26 b Fk(X)7 b Fp(\()p Fk(S)1062 1246 y Fj(q)1101 1232 y Fp(\),)32 b(where)f Fk(S)1513 1246 y Fj(q)1577 1232 y Fi(2)26 b(P)39 b Fp(\()p Fk(q)30 b Fp(=)p 1969 1163 132 4 v 26 w(1)p Fk(;)15 b(p)q Fp(\),)32 b Fk(X)7 b Fp(\()p Fk(S)2366 1246 y Fj(q)2405 1232 y Fp(\))31 b(is)f(the)i(v)m(ertex)g(set)g(of)f (the)382 1345 y(path)k Fk(S)655 1359 y Fj(q)693 1345 y Fp(.)54 b(By)36 b(supp)s(osition,)d(the)j(v)m(ertex)g Fk(x)1928 1360 y Fj(k)2004 1345 y Fi(2)c Fk(X)7 b Fp(\()p Fk(S)2270 1359 y Fj(q)2309 1345 y Fp(\))35 b(preceding)f Fk(x)2845 1359 y Fj(j)2917 1345 y Fp(in)g(the)h(path)382 1458 y Fk(S)438 1472 y Fj(q)506 1458 y Fp(is)29 b(connected)j(with)d (the)h(v)m(ertex)i Fk(x)1713 1472 y Fj(j)1780 1458 y Fp(b)m(y)e(some)h(directed)e(path)h(of)h(length)f Fk(l)d(>)e Fp(1.)523 1571 y(Let)33 b(\012\()p Fk(x)841 1586 y Fj(k)884 1571 y Fk(;)15 b(x)976 1585 y Fj(j)1013 1571 y Fp(\))33 b(b)s(e)f(a)h(path)f(of)h(the)f(maxim)m(um)g(length)g Fk(l)i Fp(\()p Fk(l)e(>)c Fp(1\),)34 b(connected)g(to)382 1693 y Fk(x)434 1708 y Fj(k)504 1693 y Fp(and)27 b Fk(x)730 1707 y Fj(j)793 1693 y Fp(in)f(the)i(graph)1318 1670 y Fk(~)1305 1693 y(G)1377 1707 y Fj(t)1406 1693 y Fp(\()p Fk(X)1523 1660 y Fh(0)1563 1693 y Fp(\).)40 b(Ob)m(viously)-8 b(,)27 b(this)f(path)h(con)m(tains)g(only)g(essen)m(tial)382 1816 y(arcs)k(of)687 1793 y Fk(~)673 1816 y(G)745 1830 y Fj(t)774 1816 y Fp(\()p Fk(X)891 1783 y Fh(0)931 1816 y Fp(\).)523 1929 y(Change)e(the)g(MPP)g Fi(P)37 b Fp(in)28 b(the)h(follo)m(wing)e(w)m(a)m(y)-8 b(.)42 b(Comp)s(ose)29 b(a)g(path)g Fk(S)2932 1896 y Fv(\003)2927 1952 y Fj(q)3000 1929 y Fp(suc)m(h)g(that)1246 2133 y Fk(X)7 b Fp(\()p Fk(S)1424 2096 y Fv(\003)1419 2156 y Fj(q)1464 2133 y Fp(\))26 b(=)f Fk(X)7 b Fp(\()p Fk(S)1794 2147 y Fj(q)1832 2133 y Fp(\))21 b Fi([)f Fk(X)7 b Fp(\(\012\()p Fk(x)2239 2148 y Fj(k)2282 2133 y Fk(;)15 b(x)2374 2147 y Fj(j)2411 2133 y Fp(\)\))p Fk(:)523 2338 y Fp(It)31 b(is)e(easy)i(to)g(see)g (that)g(the)g(v)m(ertex)g Fk(x)1852 2352 y Fj(j)1919 2338 y Fp(will)d(b)s(e)h(mark)m(ed)i(in)e(the)h(path)h Fk(S)3074 2305 y Fv(\003)3069 2360 y Fj(q)3113 2338 y Fp(.)523 2450 y(Replace)39 b(the)f(path)h Fk(S)1312 2464 y Fj(q)1388 2450 y Fi(2)g(U)47 b Fp(b)m(y)38 b(the)h(path)f Fk(S)2172 2417 y Fv(\003)2167 2473 y Fj(q)2212 2450 y Fp(.)64 b(F)-8 b(urther,)41 b(delete)e(all)e(v)m(ertices)382 2563 y(of)g Fk(S)553 2530 y Fv(\003)548 2586 y Fj(q)629 2563 y Fp(from)f(other)h(paths)g(of)f Fi(P)7 b Fp(.)61 b(Clearly)-8 b(,)38 b(then)e(there)h(will)d(b)s(e)i(a)h(new)g(MPP)f Fi(P)3305 2530 y Fv(\003)3345 2563 y Fp(,)382 2676 y(con)m(taining)d (the)i(path)f Fk(S)1265 2643 y Fv(\003)1260 2699 y Fj(q)1304 2676 y Fp(.)52 b(Other)33 b(paths)h(of)g Fi(P)2076 2643 y Fv(\003)2150 2676 y Fp(will)e(con)m(tain)i(the)g(residual)e(paths)382 2789 y(of)e Fi(P)7 b Fp(.)523 2902 y(Since)29 b(the)h(v)m(ertex)h Fk(x)1243 2916 y Fj(j)1309 2902 y Fp(is)e(mark)m(ed)h(in)e(the)i(MPP)g Fi(P)7 b Fp(,)31 b(then)e(the)h(path)g Fk(S)2977 2916 y Fj(q)3044 2902 y Fp(con)m(tains)382 3015 y(at)g(least)g(one)g(v)m (ertex)h Fk(x)1197 3029 y Fj(i)1255 3015 y Fp(preceding)d(the)i(v)m (ertex)h Fk(x)2146 3030 y Fj(k)2218 3015 y Fp(suc)m(h)f(that)g Fk(x)2671 3029 y Fj(i)2729 3015 y Fp(and)f Fk(x)2957 3029 y Fj(j)3023 3015 y Fp(are)h(inde-)382 3128 y(p)s(enden)m(t.)55 b(Therefore,)37 b(the)e(path)h Fk(S)1665 3095 y Fv(\003)1660 3150 y Fj(q)1739 3128 y Fp(con)m(tains)g(these)g(indep)s(enden)m(t)d(v) m(ertices)j(also.)382 3241 y(That)30 b(is,)g Fk(x)778 3255 y Fj(j)840 3241 y Fi(2)24 b(B)s Fp(\()p Fi(P)1093 3208 y Fv(\003)1133 3241 y Fp(\).)523 3354 y(It)30 b(is)f(easy)i(to)f (see)h(that)g(the)f(MPP)g Fi(P)1829 3321 y Fv(\003)1899 3354 y Fp(is)f(correct)i(with)e(resp)s(ect)h(to)g(the)h(mark)m(ed)382 3467 y(v)m(ertex)g Fk(x)709 3481 y Fj(j)746 3467 y Fp(.)2322 b(Q.E.D.)523 3679 y(Notice)43 b(that)f(a)g(new)f(mark)m(ed)h(v)m(ertex) h(ma)m(y)f(app)s(ear)f(in)f(the)i(same)g(path)f(as)h(a)382 3792 y(result)35 b(of)g(constructing)h(the)g(correct)g(MPP)g(with)f (resp)s(ect)g(to)i(the)f(mark)m(ed)f(v)m(ertex)382 3905 y Fk(x)434 3919 y Fj(j)470 3905 y Fp(.)1028 5023 y 3.3 setlinewidth 1206 4161 a Fc(@)1289 4244 y(@)1372 4327 y(@)1455 4411 y(@)1539 4494 y(@)1596 4551 y(@)-473 b(\000)1289 4468 y(\000)1360 4398 y(\000)1415 4787 y(B)1387 4704 y(B)1360 4621 y(B)1332 4538 y(B)1304 4455 y(B)1277 4372 y(B)1249 4289 y(B)1221 4206 y(B)1206 4162 y(B)1206 4078 y Fd(x)1443 4315 y(x)1206 4551 y(x)398 b(x)1206 4787 y(x)162 b(x)f(x)1232 4871 y Fk(x)1284 4885 y Fh(1)1468 4871 y Fk(x)1520 4885 y Fh(2)1704 4871 y Fk(x)1756 4885 y Fh(3)1066 4516 y Fk(x)1118 4530 y Fh(4)1704 4516 y Fk(x)1756 4530 y Fh(5)1468 4280 y Fk(x)1520 4294 y Fh(6)1232 4044 y Fk(x)1284 4058 y Fh(7)p 2150 4787 4 709 v 2151 4161 a Fc(@)2234 4244 y(@)2317 4327 y(@)2400 4411 y(@)2483 4494 y(@)2541 4551 y(@)2155 4158 y(@)2238 4241 y(@)2308 4311 y(@)2158 4154 y(@)2241 4237 y(@)2312 4308 y(@)2148 4165 y(@)2231 4248 y(@)2301 4318 y(@)2144 4169 y(@)2227 4252 y(@)2297 4322 y(@)2151 4551 y(\000)2234 4468 y(\000)2304 4398 y(\000)2155 4554 y(\000)2238 4471 y(\000)2308 4401 y(\000)2158 4558 y(\000)2241 4475 y(\000)2312 4405 y(\000)2148 4547 y(\000)2231 4464 y(\000)2301 4394 y(\000)2144 4544 y(\000)2227 4461 y(\000)2297 4391 y(\000)p 2386 4787 4 473 v 2360 4787 a(B)2332 4704 y(B)2304 4621 y(B)2277 4538 y(B)2249 4455 y(B)2221 4372 y(B)2194 4289 y(B)2166 4206 y(B)2151 4162 y(B)2151 4078 y Fd(x)2388 4315 y(x)2151 4551 y(x)398 b(x)2151 4787 y(x)162 b(x)f(x)2176 4871 y Fk(x)2228 4885 y Fh(1)2413 4871 y Fk(x)2465 4885 y Fh(2)2649 4871 y Fk(x)2701 4885 y Fh(3)2011 4516 y Fk(x)2063 4530 y Fh(4)2649 4516 y Fk(x)2701 4530 y Fh(5)2413 4280 y Fk(x)2465 4294 y Fh(6)2176 4044 y Fk(x)2228 4058 y Fh(7)1385 4999 y Fp(\(a\))827 b(\(b\))p 1194 4787 25 709 v 1666 4787 25 237 v 1430 4787 25 473 v 2611 4787 25 237 v 2139 4787 V 1047 5219 a(Fig.)30 b(13:)42 b(Construction)29 b(of)i(the)f(correct)i (MPP)1831 5652 y(26)p eop %%Page: 27 27 27 26 bop 523 548 a Fp(Fig.)38 b(13)24 b(\(a\))f(sho)m(ws)g(the)f(MPP)h (that)g(is)f(not)h(correct)g(with)f(resp)s(ect)g(to)i(the)f(mark)m(ed) 382 661 y(v)m(ertex)36 b Fk(x)714 675 y Fh(7)753 661 y Fp(.)55 b(The)35 b(correct)h(MPP)f(with)f(resp)s(ect)h(to)h(the)f (mark)m(ed)g(v)m(ertex)h Fk(x)3040 675 y Fh(7)3079 661 y Fp(,)h(whic)m(h)382 774 y(w)m(as)26 b(constructed)g(during)d(the)i (pro)m(ving)g(of)h(Theorem)f(10,)i(is)e(sho)m(wn)g(in)f(Fig.)39 b(13)26 b(\(b\).)382 1060 y Fl(9)135 b(Elimination)47 b(of)e(a)g(Mark)l(ed)g(V)-11 b(ertex)382 1266 y Fa(9.1)112 b(The)38 b(concept)f(of)h(a)g(standard)g(transformation)f(of)h(a)g (digraph)382 1438 y Fp(Let)31 b(us)e(come)j(bac)m(k)f(to)g(the)g (example)f(considered)f(in)g(Section)h(6.)523 1551 y(Figs.)64 b(11)39 b(\(a\))g(and)f(\(b\))g(represen)m(t)g(the)g(new)g(digraphs.)62 b(If,)40 b(in)d(the)h(transitiv)m(e)382 1664 y(closure)28 b(graph)g(of)g(an)m(y)h(of)f(these)h(digraphs,)e(w)m(e)i(\014nd)e(a)i (new)f(MPP)g Fg(arbitr)-5 b(arily)31 b Fp(then,)382 1777 y(in)h(general)h(case,)h(the)g(purp)s(ose)d(of)i(mo)s(difying)d(the)k (initial)c(digraph)h({)i(to)h(solv)m(e)f(the)382 1890 y(MPP-problem)e(for)i(the)f(TCG)g(of)h(the)g(digraph)e(\(9)i(\(a\)\))h ({)f(ma)m(y)g(b)s(e)f(lost.)47 b(T)-8 b(o)33 b(a)m(v)m(oid)382 2003 y(this,)c(in)g(the)i(TCG)e(of)i(the)f(new)g(digraph,)f(w)m(e)h(m)m (ust)h Fg(save)p Fp(,)f(as)h(the)f(initial)e(partition,)382 2116 y(all)d(engaged)j(\(thic)m(k\))f(arcs)g(of)g(TCG)f(of)h(the)g (initial)c(digraph)i(\(see,)k(for)d(example,)h(Fig.)382 2228 y(11)k(\(b\)\).)523 2341 y(Th)m(us,)26 b(w)m(e)h(come)h(to)f(the)f (concept)i(of)e(a)h Fg(standar)-5 b(d)32 b(tr)-5 b(ansformation)33 b(of)c(a)g(digr)-5 b(aph)p Fp(.)523 2454 y(Let)27 b(there)g(b)s(e)f(a)h (digraph)1447 2431 y Fk(~)1433 2454 y(G)p Fp(\()p Fk(X)1622 2421 y Fh(0)1662 2454 y Fp(\))g(and)f(let)h(some)g(MPP)g Fi(P)34 b Fp(of)2694 2431 y Fk(~)2680 2454 y(G)2752 2468 y Fj(t)2781 2454 y Fp(\()p Fk(X)2898 2421 y Fh(0)2939 2454 y Fp(\))27 b(b)s(e)f(found.)382 2567 y(W)-8 b(e)40 b(supp)s(ose)e(that)i(some)g(arcs)f(of)1664 2544 y Fk(~)1650 2567 y(G)o Fp(\()p Fk(X)1838 2534 y Fh(0)1879 2567 y Fp(\))g(are)h(mark)m(ed)f(as)h(\\considered",)i(and)c(a)382 2680 y(mark)m(ed)30 b(v)m(ertex)i Fk(x)1033 2694 y Fj(j)1100 2680 y Fp(is)d(found)g(in)g(the)i(TCG)1955 2657 y Fk(~)1942 2680 y(G)2014 2694 y Fj(t)2043 2680 y Fp(\()p Fk(X)2160 2647 y Fh(0)2200 2680 y Fp(\).)523 2803 y(W)-8 b(e)40 b(will)c(call)i(a)h(transformation)f(of)h(the)g(digraph)2387 2780 y Fk(~)2374 2803 y(G)o Fp(\()p Fk(X)2562 2770 y Fh(0)2603 2803 y Fp(\))g(standard)e(when)h(it)382 2916 y(consists)30 b(in)f(realization)h(of)g(the)h(follo)m(wing)d(steps.)382 3101 y Fg(Step)33 b(1)46 b Fp(Find)29 b(the)h(leading)g(arc)g Fk(x)1565 3116 y Fj(k)1633 3101 y Fk(>)25 b(x)1781 3115 y Fj(j)1848 3101 y Fp(in)k(the)i(TCG)2357 3078 y Fk(~)2344 3101 y(G)2416 3115 y Fj(t)2445 3101 y Fp(\()p Fk(X)2562 3068 y Fh(0)2602 3101 y Fp(\).)382 3237 y Fg(Step)i(2)46 b Fp(Mark)31 b(the)f(found)f(arc)i(as)g(\\considered.")382 3374 y Fg(Step)i(3)46 b Fp(Remo)m(v)m(e)37 b Fq(all)d Fp(arcs,)j(mark)m(ed)e(as)g(\\considered")f(in)2493 3351 y Fk(~)2480 3374 y(G)o Fp(\()p Fk(X)2668 3341 y Fh(0)2708 3374 y Fp(\).)55 b(As)34 b(the)h(result,)609 3487 y(w)m(e)c(will)d(ha)m (v)m(e)j(a)g(new)f(digraph)1734 3464 y Fk(~)1721 3487 y(G)1793 3454 y Fv(0)1815 3487 y Fp(\()p Fk(X)1932 3454 y Fh(0)1973 3487 y Fp(\).)382 3623 y Fg(Step)j(4)46 b Fp(Construct)30 b(the)h(TCG)1504 3600 y Fk(~)1490 3623 y(G)1562 3590 y Fv(0)1562 3646 y Fj(t)1592 3623 y Fp(\()p Fk(X)1709 3590 y Fh(0)1749 3623 y Fp(\).)382 3760 y Fg(Step)i(5)46 b Fp(Keep)36 b(all)e(thic)m(k)i(arcs)g(of)g(MPP)g Fi(P)43 b Fp(of)2052 3737 y Fk(~)2038 3760 y(G)2110 3774 y Fj(t)2139 3760 y Fp(\()p Fk(X)2256 3727 y Fh(0)2297 3760 y Fp(\))36 b(\(except)h(the)f(arc)g Fk(x)3065 3775 y Fj(k)3142 3760 y Fk(>)e(x)3299 3774 y Fj(j)3335 3760 y Fp(\))609 3883 y(thic)m(k)k(for)1005 3860 y Fk(~)992 3883 y(G)1064 3850 y Fv(0)1064 3905 y Fj(t)1093 3883 y Fp(\()p Fk(X)1210 3850 y Fh(0)1250 3883 y Fp(\))h(and)f(\014nd)e(a)j(new)f(MPP)h(in)e (the)h(TCG)2756 3860 y Fk(~)2742 3883 y(G)2814 3850 y Fv(0)2814 3905 y Fj(t)2843 3883 y Fp(\()p Fk(X)2960 3850 y Fh(0)3001 3883 y Fp(\))g(\(in)g(this)609 3996 y(case)32 b(w)m(e)e(c)m(ho)s(ose)i(only)d(essen)m(tial)h(arcs\).)523 4181 y(It)f(is)f(easy)i(to)g(see)f(if)f(in)g(the)h(example)g (considered)f(ab)s(o)m(v)m(e)i(w)m(e)g(execute)g(the)f(stan-)382 4294 y(dard)f(transformation)g(of)h(the)g(digraph)e(9,)j(\(a\))g(then)f (w)m(e)g(will)e(obtain)h(an)h(MPP)-8 b(,)29 b(arcs)382 4406 y(of)34 b(whic)m(h)e(are)i(represen)m(ted)g(in)e(Fig.)51 b(11.)g(Ob)m(viously)-8 b(,)34 b(the)f(obtained)h(MPP)f(will)e(b)s(e) 382 4519 y(the)f(solution)f(to)j(the)e(MPP-problem)f(that)i(w)m(e)g (are)g(seeking.)523 4632 y(T)-8 b(o)30 b(a)m(v)m(oid)g(am)m(biguit)m(y) -8 b(,)30 b(w)m(e)g(ha)m(v)m(e)h(already)e(determined)f(the)i(order)f (of)h(considera-)382 4745 y(tion)g(of)h(w)m(orking)f(tables)h(for)g (the)g(ordered)f(set)h(\(see)h(Section)f(2\).)43 b(In)30 b(this)f(table)i(it)g(is)382 4858 y(mean)m(t)d(that)g(the)f(elemen)m (ts)h(of)f(the)g(ordered)g(set)h(are)f(n)m(um)m(b)s(ered)f(in)g(some)h (w)m(a)m(y)-8 b(.)41 b(This)382 4971 y(n)m(um)m(b)s(ering)26 b(actually)i(determines)g(the)h(priorit)m(y)e(of)h(consideration)g(of)h (the)f(elemen)m(ts)382 5084 y(of)22 b(the)g(ordered)f(set.)38 b(The)21 b(elemen)m(t)i(ha)m(ving)e(a)h(smaller)e(n)m(um)m(b)s(er)g(is) h(considered)g(earlier)382 5197 y(than)28 b(the)g(elemen)m(t)g(with)f (a)h(large)g(n)m(um)m(b)s(er,)g(that)g(is,)g(the)g(elemen)m(t)g(with)f (the)h(smaller)382 5310 y(n)m(um)m(b)s(er)h(has)h(a)h(greater)g (priorit)m(y)e(than)h(the)h(elemen)m(t)g(with)e(the)h(greater)i(n)m(um) m(b)s(er.)1831 5652 y(27)p eop %%Page: 28 28 28 27 bop 523 548 a Fp(F)-8 b(or)32 b(the)g(same)g(purp)s(ose,)f (during)e(realization)i(of)h(standard)f(transformation,)g(it)382 661 y(is)39 b(exp)s(edien)m(t)h(to)h(determine)f(priorities)e(of)i(v)m (ertices)h(of)g(the)g(VS-digraph)3095 638 y Fk(~)3081 661 y(G)p Fp(\()p Fk(X)3270 628 y Fh(0)3310 661 y Fp(\),)382 774 y(since)36 b(these)h(v)m(ertices)h(are)f(the)g(elemen)m(ts)g(of)g (the)g(ordered)f(set)h({)g(of)g(the)g(transitiv)m(e)382 887 y(closure)28 b(graph)954 864 y Fk(~)940 887 y(G)1012 901 y Fj(t)1041 887 y Fp(\()p Fk(X)1158 854 y Fh(0)1199 887 y Fp(\).)40 b(W)-8 b(e)30 b(will)c(supp)s(ose)h(that)j(eac)m(h)g(v) m(ertex)f(of)g(the)g(VS-digraph,)382 1000 y(b)s(elonging)e(to)j(the)g Fk(k)1105 1014 y Fh(1)1144 1000 y Fp(-th)g(la)m(y)m(er,)g(has)f(a)g (greater)i(priorit)m(y)c(than)i(a)h(v)m(ertex,)g(b)s(elonging)382 1112 y(to)36 b(the)g Fk(k)707 1126 y Fh(2)747 1112 y Fp(-th)f(la)m(y)m(er,)j(if)d Fk(k)1286 1126 y Fh(1)1359 1112 y Fk(<)f(k)1511 1126 y Fh(2)1551 1112 y Fp(.)56 b(Besides,)37 b(the)e(v)m(ertices)i(of)e(ev)m(ery)i(la)m(y)m(er)f(also) f(are)382 1225 y(n)m(um)m(b)s(ered,)30 b(that)h(is,)g(some)g(priorit)m (y)e(is)h(selected)i(for)e(eac)m(h)i(of)f(v)m(ertices)h(of)f(the)g (same)382 1338 y(la)m(y)m(er.)523 1451 y(The)d(determined)g(priorit)m (y)f(of)i(v)m(ertices)h(of)f(a)g(VS-digraph)e(actually)i(determines)382 1564 y(the)f(order)f(for)h(searc)m(hing)g(for)g(an)f(alternating)h (path)g(during)d(construction)j(of)g(a)g(new)382 1677 y(MPP)-8 b(.)382 1890 y Fq(Theorem)34 b(11)46 b Fg(The)39 b(standar)-5 b(d)42 b(tr)-5 b(ansformation)42 b(r)-5 b(e)g(quir)g(es)40 b(an)f Fk(O)s Fp(\()p Fk(n)2810 1857 y Fh(3)2849 1890 y Fp(\))g Fg(exp)-5 b(enditur)g(e)382 2002 y(of)33 b(the)g(time)g(units.)523 2215 y Fp(Eac)m(h)e(step)g(of)g (the)f(standard)g(transformation)g(is)g(p)s(erformed)f(once,)i(and)f (there-)382 2328 y(fore,)24 b(the)f(running)c(time)j(of)g(this)f (transformation)h(is)f(determined)g(b)m(y)h(the)g(time)g(of)g(the)382 2441 y(longest)32 b(of)h(them)f({)g(this)f(is)h(Step)g(4)g({)h(the)f (construction)g(of)g(the)h(transitiv)m(e)e(closure)382 2554 y(graph.)2458 b(Q.E.D.)382 2797 y Fa(9.2)112 b(An)37 b(auxiliary)g(digraph)382 2969 y Fp(Let)32 b(us)e(consider)h(the)g (general)h(case,)h(i.e.,)e(w)m(e)h(will)d(consider)h(what)h (information)f(w)m(e)382 3082 y(can)j(extract)i(as)e(the)h(result)e(of) h(realization)g(of)g(the)g(standard)f(transformation)h(of)g(a)382 3195 y(digraph.)523 3308 y(Let)710 3285 y Fk(~)697 3308 y(G)o Fp(\()p Fk(X)885 3275 y Fh(0)925 3308 y Fp(\))42 b(b)s(e)f(a)g(VS-digraph,)i(and)d(let)2088 3285 y Fk(~)2074 3308 y(G)2146 3322 y Fj(t)2176 3308 y Fp(\()p Fk(X)2293 3275 y Fh(0)2333 3308 y Fp(\))h(b)s(e)g(its)g(transitiv)m(e)f(closure) 382 3420 y(graph.)523 3533 y(F)-8 b(urther,)40 b(let)f Fi(P)46 b Fp(b)s(e)38 b(the)h(MPP)f(of)1804 3510 y Fk(~)1791 3533 y(G)1863 3547 y Fj(t)1892 3533 y Fp(\()p Fk(X)2009 3500 y Fh(0)2049 3533 y Fp(\))h(suc)m(h)f(that)h Fi(B)s Fp(\()p Fi(P)7 b Fp(\))40 b Fi(6)p Fp(=)e Fi(\013)p Fp(.)65 b(W)-8 b(e)40 b(also)382 3646 y(supp)s(ose)32 b(that)j(the)f(paths)f (of)h Fi(P)42 b Fp(are)34 b(ordered)f(b)m(y)h(n)m(um)m(b)s(ering)e(v)m (ertices)i(of)g Fk(X)3145 3613 y Fh(0)3219 3646 y Fp(\(see)382 3759 y(Theorem)c(1\).)523 3872 y(Clearly)-8 b(,)37 b(the)g(sense)g(of)f (the)h(standard)f(transformation)g(consists)g(in)f(searc)m(hing)382 3985 y(for)d(a)h(new)f(MPP)g(of)h(the)f(TCG)g(for)1683 3962 y Fk(~)1670 3985 y(G)o Fp(\()p Fk(X)1858 3952 y Fh(0)1898 3985 y Fp(\))h(b)m(y)f(a)h(construction)f(of)h(a)f(new)g (digraph)382 4098 y(in)39 b(whic)m(h)g(some)h(\014ctitious)f(arc)i(is)e (eliminated,)i(t)m(w)m(o)g(terminal)e(v)m(ertices)i(of)f(whic)m(h)382 4211 y(b)s(elong)30 b(to)h(the)h(same)f(path)g(of)g(MPP)g Fi(P)7 b Fp(.)43 b(The)30 b(new)g(digraph)2585 4188 y Fk(~)2572 4211 y(G)2644 4178 y Fv(0)2666 4211 y Fp(\()p Fk(X)2783 4178 y Fh(0)2824 4211 y Fp(\),)h(obtained)f(as)382 4324 y(the)g(result)g(of)g(the)h(standard)e(transformation,)h(w)m(e)h (will)d(call)i Fg(auxiliary)p Fp(.)523 4536 y(Let)701 4513 y Fk(~)688 4536 y(G)760 4503 y Fv(0)782 4536 y Fp(\()p Fk(X)899 4503 y Fh(0)940 4536 y Fp(\))i(b)s(e)g(the)g(auxiliary)e (digraph)g(obtained)i(as)g(the)g(result)g(of)g(the)g(stan-)382 4649 y(dard)21 b(transformation)h(of)h(the)g(digraph)1782 4626 y Fk(~)1769 4649 y(G)o Fp(\()p Fk(X)1957 4616 y Fh(0)1997 4649 y Fp(\))g(with)e(resp)s(ect)i(to)g(the)g(mark)m(ed)f(v)m (ertex)382 4762 y Fk(x)434 4776 y Fj(j)470 4762 y Fp(.)523 4875 y(W)-8 b(e)37 b(will)c(construct)j(the)g(TCG)g(of)g(the)g (auxiliary)d(digraph)h(and)h(\014nd)f(its)i(MPP)382 4988 y Fi(P)452 4955 y Fv(0)506 4988 y Fp(as)31 b(indicated)e(ab)s(o)m(v)m (e.)42 b(Ob)m(viously)-8 b(,)29 b(in)g(general)h(case)i(w)m(e)e(will)e (ha)m(v)m(e:)1452 5192 y Fk(C)7 b(ar)s(d)p Fp(\()p Fi(P)1768 5155 y Fv(0)1792 5192 y Fp(\))26 b Fi(\025)f Fk(C)7 b(ar)s(d)p Fp(\()p Fi(P)g Fp(\))523 5396 y(Th)m(us,)30 b(w)m(e)g(ma)m(y)h (consider)f(t)m(w)m(o)h(cases:)1831 5652 y(28)p eop %%Page: 29 29 29 28 bop 518 548 a Fi(\017)46 b Fq(Case)35 b(1.)40 b Fk(C)7 b(ar)s(d)p Fp(\()p Fi(P)1297 515 y Fv(0)1321 548 y Fp(\))26 b(=)f Fk(C)7 b(ar)s(d)p Fp(\()p Fi(P)g Fp(\).)518 736 y Fi(\017)46 b Fq(Case)35 b(2.)40 b Fk(C)7 b(ar)s(d)p Fp(\()p Fi(P)1297 703 y Fv(0)1321 736 y Fp(\))26 b Fk(>)f(C)7 b(ar)s(d)p Fp(\()p Fi(P)g Fp(\).)382 948 y Fq(Claim)33 b(1)46 b Fg(If)34 b Fk(C)7 b(ar)s(d)p Fp(\()p Fi(P)1199 915 y Fv(0)1222 948 y Fp(\))28 b(=)f Fk(C)7 b(ar)s(d)p Fp(\()p Fi(P)g Fp(\))p Fg(,)35 b(then)f(an)h(MPP)d Fk(C)7 b(ar)s(d)p Fp(\()p Fi(P)2686 915 y Fv(0)2710 948 y Fp(\))34 b Fg(wil)5 b(l)34 b(also)h(b)-5 b(e)34 b(the)382 1061 y(MPP)d(of)740 1038 y Fk(~)726 1061 y(G)798 1075 y Fj(t)827 1061 y Fp(\()p Fk(X)944 1028 y Fh(0)984 1061 y Fp(\))p Fg(.)523 1274 y Fp(It)k(is)g(ob)m(vious)f(since)h(all)f(arcs)i(of)f (the)g(auxiliary)e(digraph)h(are)h(also)h(arcs)f(of)g(the)382 1386 y(initial)28 b(digraph.)2118 b(Q.E.D.)382 1599 y Fq(Claim)33 b(2)46 b Fg(If)40 b Fk(C)7 b(ar)s(d)p Fp(\()p Fi(P)1205 1566 y Fv(0)1228 1599 y Fp(\))39 b(=)f Fk(C)7 b(ar)s(d)p Fp(\()p Fi(P)g Fp(\))p Fg(,)42 b(then)f(an)f(MPP)f Fk(C)7 b(ar)s(d)p Fp(\()p Fi(P)2740 1566 y Fv(0)2763 1599 y Fp(\))40 b Fg(c)-5 b(ontains)42 b(only)382 1712 y(essential)33 b(ar)-5 b(cs.)523 1924 y Fp(Since)35 b(the)h(auxiliary)e (digraph)g(is)h(a)h(VS-digraph,)g(it)g(therefore)g(\(see)h(Theorem)382 2037 y(5\))31 b(has)f(an)g(MPP)-8 b(,)31 b(con)m(taining)f(only)g (essen)m(tial)g(arcs.)871 b(Q.E.D.)523 2250 y(If)31 b(Case)g(1)h(o)s (ccurs,)g(w)m(e)f(distinguish)d(t)m(w)m(o)33 b(particular)c(cases)j (for)f(the)h(MPP)f Fi(P)3242 2217 y Fv(0)3297 2250 y Fp(of)395 2340 y Fk(~)382 2363 y(G)454 2377 y Fj(t)483 2363 y Fp(\()p Fk(X)600 2330 y Fh(0)640 2363 y Fp(\):)518 2575 y Fi(\017)46 b Fq(Case)35 b(1.1.)41 b Fp(The)29 b(v)m(ertex)j Fk(x)1577 2589 y Fj(j)1644 2575 y Fp(is)d(not)i(mark)m (ed.)518 2763 y Fi(\017)46 b Fq(Case)35 b(1.2.)41 b Fp(The)29 b(v)m(ertex)j Fk(x)1577 2777 y Fj(j)1644 2763 y Fp(is)d(mark)m(ed)i (again.)523 2976 y(Clearly)-8 b(,)24 b(if)e(Case)i(1.1)g(o)s(ccurs,)h (then)e(the)g(purp)s(ose)f(of)i(constructing)e(the)i(auxiliary)382 3088 y(digraph)30 b(is)h(attained)h({)g(w)m(e)g(ha)m(v)m(e)h(obtained)f (the)g(MPP)g(of)2484 3065 y Fk(~)2471 3088 y(G)2543 3102 y Fj(t)2572 3088 y Fp(\()p Fk(X)2689 3055 y Fh(0)2729 3088 y Fp(\))g(not)g(con)m(taining)382 3201 y(the)e(v)m(ertex)i Fk(x)866 3215 y Fj(j)933 3201 y Fp(as)e(mark)m(ed.)523 3314 y(If)35 b(Case)g(1.2)h(o)s(ccurs)f(then,)h(ob)m(viously)-8 b(,)36 b(w)m(e)g(ma)m(y)f(again)h(execute)g(the)f(standard)382 3427 y(transformation)28 b(of)h(the)g(digraph)1602 3404 y Fk(~)1589 3427 y(G)o Fp(\()p Fk(X)1777 3394 y Fh(0)1818 3427 y Fp(\))g(with)e(resp)s(ect)i(to)g(the)g(mark)m(ed)g(v)m(ertex)h Fk(x)3309 3441 y Fj(j)3345 3427 y Fp(.)382 3671 y Fa(9.3)112 b(An)37 b(analyzing)h(digraph)382 3842 y Fp(Let)31 b(there)f(b)s(e)g (Case)h(2.)523 3955 y(W)-8 b(e)31 b(will)d(\014nd)h(the)h(general)g(an) m(tipath)g Fi(U)1934 3922 y Fv(0)1988 3955 y Fp(\(see)h(Section)f(4\))h (in)e(the)h(MPP)h Fi(P)3169 3922 y Fv(0)3193 3955 y Fp(.)40 b(By)382 4068 y(the)30 b(condition)f(w)m(e)i(ha)m(v)m(e)h(in)d(this)g (case:)42 b Fk(C)7 b(ar)s(d)p Fp(\()p Fi(U)2104 4035 y Fv(0)2127 4068 y Fp(\))26 b Fk(>)f(C)7 b(ar)s(dX)2577 4035 y Fh(0)2616 4068 y Fp(.)523 4181 y(Tw)m(o)31 b(particular)d(cases) k(are)e(p)s(ossible:)518 4393 y Fi(\017)46 b Fq(Case)51 b(2.1.)85 b Fp(The)44 b(an)m(tipath)h Fi(U)1771 4360 y Fv(0)1839 4393 y Fp(con)m(tains)g(only)f(indep)s(enden)m(t)f(v)m (ertices)i(of)623 4483 y Fk(~)609 4506 y(G)p Fp(\()p Fk(X)798 4473 y Fh(0)838 4506 y Fp(\).)518 4694 y Fi(\017)h Fq(Case)j(2.2.)78 b Fp(The)43 b(an)m(tipath)g Fi(U)1759 4661 y Fv(0)1825 4694 y Fp(con)m(tains)g(at)g(least)g(one)h(pair)d(of)i (adjacen)m(t)609 4807 y(v)m(ertices)31 b(of)1055 4784 y Fk(~)1042 4807 y(G)o Fp(\()p Fk(X)1230 4774 y Fh(0)1270 4807 y Fp(\).)523 5019 y(If)j(Case)h(2.1)g(o)s(ccurs,)g(w)m(e)g(can)g (construct)g(a)f(new)g(acyclic)h(digraph,)f(ha)m(ving)f(the)382 5132 y(initiating)e(set)j(of)f(a)h(greater)g(size)f(than)h(the)f (curren)m(t)g(digraph)f(\(see)i(Section)f(4\).)50 b(In)382 5245 y(this)21 b(case,)j(the)e(pro)s(cessing)f(of)h(the)g(MPP-problem)f (should)f(b)s(e)h(stop)s(ed,)i(all)e(arc)h(marks)382 5358 y(should)e(b)s(e)h(remo)m(v)m(ed)i(and)f(a)g(new)g(VS-digraph,)g (whose)g(initiating)e(set)i(con)m(tains)g(more)1831 5652 y(29)p eop %%Page: 30 30 30 29 bop 382 548 a Fp(v)m(ertices,)32 b(should)d(b)s(e)h(constructed.) 42 b(Ob)m(viously)-8 b(,)30 b(w)m(e)h(will)e(solv)m(e)i(the)g (MPP-problem)382 661 y(for)f(the)h(TCG)f(of)g(the)h(new)f(VS-digraph)e (then.)523 774 y(Let)j(Case)g(2.2)g(o)s(ccurs.)523 887 y(In)j(the)h(maxim)m(um)e(an)m(tipath)i Fi(U)1667 854 y Fv(0)1690 887 y Fp(,)h(\014nd)d(pairs)g(of)i(adjacen)m(t)h(v)m (ertices)g(of)3095 864 y Fk(~)3081 887 y(G)p Fp(\()p Fk(X)3270 854 y Fh(0)3310 887 y Fp(\).)382 1000 y(Let)f Fk(S)605 1014 y Fh(1)675 1000 y Fi(\032)c(U)843 967 y Fv(0)901 1000 y Fp(b)s(e)i(the)h(set)h(of)f(these)h(v)m(ertices.)52 b(A)35 b(directed)e(subgraph,)h(induced)e(b)m(y)382 1112 y(v)m(ertices)25 b(of)g(the)f(set)h Fk(S)1145 1126 y Fh(1)1184 1112 y Fp(,)h(w)m(e)f(designate)g(as)1876 1090 y Fk(~)1862 1112 y(G)p Fp(\()p Fk(S)2025 1126 y Fh(1)2064 1112 y Fp(\).)40 b(In)23 b(the)i(subgraph)2824 1090 y Fk(~)2811 1112 y(G)o Fp(\()p Fk(S)2973 1126 y Fh(1)3013 1112 y Fp(\),)h(\014nd)d(an)382 1225 y(MIS)33 b Fk(U)654 1192 y Fv(\003)693 1225 y Fp(,)h(consisting)e(of)h(the)g(v)m(ertices)h (of)f(the)h Fg(maximum)i(r)-5 b(ank)p Fp(.)50 b(It)33 b(is)f(not)i(di\016cult)382 1338 y(to)42 b(see)f(that)h(the)f(v)m (ertices)g Fk(U)1447 1305 y Fv(\003)1528 1338 y Fp(together)h(with)e (the)h(adjacen)m(t)h(v)m(ertices)f(of)h Fi(U)3176 1305 y Fv(0)3226 1338 y Fi(n)28 b Fk(U)382 1451 y Fp(determine)h(arcs)i(of) 1111 1428 y Fk(~)1098 1451 y(G)o Fp(\()p Fk(X)1286 1418 y Fh(0)1326 1451 y Fp(\))g(mark)m(ed)f(as)h(\\considered.")523 1564 y(Construct)40 b(a)g(new)f(digraph)1596 1541 y Fk(~)1582 1564 y(G)1654 1531 y Fv(\003)1693 1564 y Fp(\()p Fk(X)1810 1531 y Fh(0)1851 1564 y Fp(\))h(reorien)m(ting)f Fq(all)2555 1541 y Fk(~)2542 1564 y(G)o Fp(\()p Fk(X)2730 1531 y Fh(0)2770 1564 y Fp(\))i(arcs)f(that)g(are)382 1677 y(incoming)24 b(in)m(to)i(the)g(v)m(ertices)g(of)g(the)g(set)h Fk(U)1883 1644 y Fv(\003)1948 1677 y Fp(and)e(mark)m(ed)h(as)g(\\considered".)39 b(Notice)382 1790 y(that)32 b(some)g(initial)d(v)m(ertices)j(of)g(some) g(of)f(these)h(arcs)g(can)g(b)s(e)f(not)g(b)s(elonging)f(to)i(the)382 1903 y(set)f Fk(S)580 1917 y Fh(1)619 1903 y Fp(.)41 b(The)30 b(obtained)f(digraph)g(w)m(e)i(will)c(call)j Fg(analyzing)p Fp(.)382 2096 y Fq(Theorem)k(12)46 b Fg(The)33 b(analyzing)h(digr)-5 b(aph)1899 2074 y Fk(~)1885 2096 y(G)1957 2063 y Fv(\003)1996 2096 y Fp(\()p Fk(X)2113 2063 y Fh(0)2154 2096 y Fp(\))32 b Fg(b)-5 b(elongs)34 b(to)f(the)g(set)g Fk(D)s Fp(\()p Fk(G)p Fp(\))p Fg(.)523 2290 y Fp(Supp)s(ose)d(that)1093 2267 y Fk(~)1079 2290 y(G)1151 2257 y Fv(\003)1190 2290 y Fp(\()p Fk(X)1307 2257 y Fh(0)1347 2290 y Fp(\))i(do)s(es)f(not)h(b)s(elong)f(to)h Fk(D)s Fp(\()p Fk(G)p Fp(\).)45 b(This)30 b(can)h(b)s(e)g(only)g(if,)g (as)382 2403 y(a)g(result)e(of)h(reorien)m(tation)h(of)f(arcs)h(of)f (the)h(initial)c(digraph)2514 2380 y Fk(~)2500 2403 y(G)o Fp(\()p Fk(X)2688 2370 y Fh(0)2729 2403 y Fp(\),)k(the)f(digraph)f(is) 382 2516 y(cyclic.)40 b(Suc)m(h)27 b(a)i(digraph)e(can)h(b)s(e)g (obtained)g(only)f(when)g(the)i(terminal)e(v)m(ertices)i(of)f(a)382 2629 y(reorien)m(ted)i(arc)h(are)g(connected)g(b)m(y)f(some)h(directed) f(path)g(of)2569 2606 y Fk(~)2555 2629 y(G)p Fp(\()p Fk(X)2744 2596 y Fh(0)2784 2629 y Fp(\).)523 2742 y(But)g(it)g(is)f (imp)s(ossible,)e(since)i(if)g(an)h(arc)g(of)g(an)g(MPP)-8 b(,)31 b(not)f(correct)h(with)e(resp)s(ect)382 2855 y(to)23 b(a)g(mark)m(ed)g(v)m(ertex)h Fk(x)1189 2869 y Fj(j)1225 2855 y Fp(,)h(is)c(mark)m(ed)i(as)g(\\considered",)h(then)e(an)h(MPP)-8 b(,)23 b(correct)h(with)382 2968 y(resp)s(ect)32 b(to)h(the)f(mark)m (ed)h(v)m(ertex)g Fk(x)1621 2982 y Fj(j)1658 2968 y Fp(,)f(had)g(to)h (b)s(e)e(constructed)i(to)s(o,)g(and)f(hence)g(the)382 3090 y(arc)i(incoming)e(to)j Fk(x)1096 3104 y Fj(j)1166 3090 y Fp(and)e(b)s(elonging)f(to)i(the)g(directed)f(path)h(of)2723 3067 y Fk(~)2709 3090 y(G)p Fp(\()p Fk(X)2898 3057 y Fh(0)2938 3090 y Fp(\))g(indicated)382 3203 y(ab)s(o)m(v)m(e,)e(is)d (mark)m(ed)i(as)f(\\considered".)1371 b(Q.E.D.)523 3416 y(W)-8 b(e)39 b(assume)e(that)h(the)g(priorities)d(of)j(the)g(v)m (ertices)g(of)g(the)g(analyzing)e(digraph)382 3529 y(remain)29 b(the)i(same)g(as)f(they)h(w)m(ere)g(in)e(the)h(initial)e(digraph)2468 3506 y Fk(~)2454 3529 y(G)p Fp(\()p Fk(X)2643 3496 y Fh(0)2683 3529 y Fp(\).)523 3642 y(F)-8 b(urther,)37 b(construct)e(the)h(TCG)1702 3619 y Fk(~)1689 3642 y(G)1761 3609 y Fv(\003)1761 3664 y Fj(t)1800 3642 y Fp(\()p Fk(X)1917 3609 y Fh(0)1957 3642 y Fp(\))f(of)h(the)f(analyzing)g(digraph)f(and)g (will)382 3755 y(its)c(MPP)g Fi(P)816 3722 y Fv(\003)856 3755 y Fp(.)41 b(Tw)m(o)30 b(results)g(of)g(the)h(p)s(erformed)d (transformation)i(are)h(p)s(ossible:)518 3948 y Fi(\017)46 b Fq(Case)35 b(2.2.1.)41 b Fp(The)30 b(an)m(tipath)g Fi(U)1763 3915 y Fv(\003)1833 3948 y Fp(of)1950 3925 y Fk(~)1936 3948 y(G)2008 3915 y Fv(\003)2008 3971 y Fj(t)2047 3948 y Fp(\()p Fk(X)2164 3915 y Fh(0)2204 3948 y Fp(\))h(ob)s(eys)f(the)g(relation)1522 4133 y Fk(C)7 b(ar)s(d)p Fp(\()p Fi(U)1834 4096 y Fv(\003)1873 4133 y Fp(\))26 b Fk(>)f(C)7 b(ar)s(d)p Fp(\()p Fk(X)2358 4096 y Fh(0)2397 4133 y Fp(\))p Fk(:)518 4353 y Fi(\017)46 b Fq(Case)35 b(2.2.2.)41 b Fp(The)30 b(an)m(tipath)g Fi(U)1763 4320 y Fv(0)1816 4353 y Fp(ob)s(eys)g(the)h(relation)1522 4538 y Fk(C)7 b(ar)s(d)p Fp(\()p Fi(U)1834 4501 y Fv(\003)1873 4538 y Fp(\))26 b(=)f Fk(C)7 b(ar)s(d)p Fp(\()p Fk(X)2358 4501 y Fh(0)2397 4538 y Fp(\))p Fk(:)523 4758 y Fp(In)29 b(Case)i(2.2.1)g(w)m(e)g(op)s(erate)f(as)h(w)m(e)f(did)e(in)h(Case)h (2.1,)i(that)e(is,)g(w)m(e)g(create)i(a)e(new)382 4871 y(VS-digraph,)36 b(ha)m(ving)f(the)h(initiating)d(set)k(of)f(a)g (greater)h(size)e(than)h(this)f(set)h(in)e(the)382 4984 y(digraph)729 4961 y Fk(~)715 4984 y(G)p Fp(\()p Fk(X)904 4951 y Fh(0)944 4984 y Fp(\).)41 b(All)29 b(marks)h(of)h(arcs)f(are)h (eliminated.)523 5097 y(Consider)e(Case)h(2.2.2.)382 5290 y Fq(Claim)j(3)46 b Fg(If)g(Case)h(2.2.2)h(o)-5 b(c)g(curs,)51 b(the)c(MPP)e Fi(P)2183 5257 y Fv(\003)2270 5290 y Fg(c)-5 b(ontains)48 b(an)f(ar)-5 b(c)47 b(marke)-5 b(d)48 b(as)382 5403 y(\\c)-5 b(onsider)g(e)g(d".)1831 5652 y Fp(30)p eop %%Page: 31 31 31 30 bop 523 548 a Fp(The)40 b(opp)s(osite)g(means)g(Case)h(1)f(for)h (the)f(auxiliary)e(digraph.)69 b(Con)m(tradiction.)382 661 y(Q.E.D.)382 973 y Fq(Theorem)34 b(13)46 b Fg(L)-5 b(et)1152 950 y Fk(~)1139 973 y(G)1211 940 y Fv(\003)1249 973 y Fp(\()p Fk(X)1366 940 y Fh(0)1407 973 y Fp(\))38 b Fg(b)-5 b(e)39 b(an)g(analyzing)g(digr)-5 b(aph)41 b(and)e Fi(P)2721 940 y Fv(\003)2800 973 y Fg(b)-5 b(e)38 b(an)h(MPP)e(of)395 1063 y Fk(~)382 1086 y(G)454 1053 y Fv(\003)454 1108 y Fj(t)493 1086 y Fp(\()p Fk(X)610 1053 y Fh(0)650 1086 y Fp(\))32 b Fg(such)g(that)g Fk(C)7 b(ar)s(d)p Fp(\()p Fi(P)g Fp(\))26 b(=)f Fk(C)7 b(ar)s(d)p Fp(\()p Fi(P)1896 1053 y Fv(\003)1936 1086 y Fp(\))p Fg(.)42 b(Then)32 b(the)g(digr)-5 b(aph)2761 1063 y Fk(~)2748 1086 y(G)o Fp(\()p Fk(X)2936 1053 y Fh(0)2976 1086 y Fp(\))32 b Fg(c)-5 b(ontains)382 1199 y(an)30 b(o)-5 b(dd)32 b(cycle)d Fk(C)961 1214 y Fh(2)p Fj(k)r Fh(+1)1159 1199 y Fg(\()p Fk(k)g Fi(\025)24 b Fp(2)p Fg(\))31 b(that)g(is)f(r)-5 b(estricte)g(d)5 b(ly)31 b(str)-5 b(etche)g(d)32 b(on)e(p)-5 b(aths)32 b(of)e(MPP)e(of)395 1289 y Fk(~)382 1312 y(G)454 1326 y Fj(t)483 1312 y Fp(\()p Fk(X)600 1279 y Fh(0)640 1312 y Fp(\))p Fg(.)523 1524 y Fp(Let)j(the)f(conditions)f(of)i (Theorem)f(13)h(b)s(e)f(satis\014ed.)523 1637 y(According)k(to)h(3,)g (the)g(MPP)f Fi(P)1651 1604 y Fv(\003)1725 1637 y Fp(con)m(tains)g(an)g (arc,)i(mark)m(ed)e(as)h(\\considered".)382 1750 y(Let)j(it)f(b)s(e)f (an)i(arc)f Fk(x)1124 1764 y Fj(j)1198 1750 y Fk(>)f(x)1357 1765 y Fj(k)1400 1750 y Fp(.)62 b(Then)36 b(the)h(MPP)h Fi(P)45 b Fp(con)m(tains)37 b(the)h(arc)g Fk(x)2985 1765 y Fj(k)3064 1750 y Fk(>)f(x)3224 1764 y Fj(j)3297 1750 y Fp(of)395 1850 y Fk(~)382 1873 y(G)o Fp(\()p Fk(X)570 1840 y Fh(0)611 1873 y Fp(\).)523 1986 y(Tigh)m(ten)25 b(the)h(arc)h(\()p Fk(x)1246 2001 y Fj(k)1289 1986 y Fk(;)15 b(x)1381 2000 y Fj(j)1418 1986 y Fp(\))26 b(in)e(the)i(digraph) 2075 1963 y Fk(~)2061 1986 y(G)p Fp(\()p Fk(X)2250 1953 y Fh(0)2290 1986 y Fp(\))g(and)g(the)g(arc)g(\()p Fk(x)2910 2000 y Fj(j)2947 1986 y Fk(;)15 b(x)3039 2001 y Fj(k)3082 1986 y Fp(\))26 b(in)e(the)382 2109 y(digraph)726 2086 y Fk(~)712 2109 y(G)784 2076 y Fv(\003)823 2109 y Fp(\()p Fk(X)940 2076 y Fh(0)981 2109 y Fp(\))j(corresp)s(ondingly)e(at)i(one)h (v)m(ertex)g Fk(x)2293 2124 y Fj(k)r(;j)2388 2109 y Fp(.)40 b(As)27 b(the)g(result)f(w)m(e)i(obtain)382 2233 y(the)k(same)g (digraph)1121 2210 y Fk(~)1105 2233 y(H)7 b Fp(\()p Fk(X)1305 2200 y Fh(0)1345 2233 y Fp(\))32 b(in)f(b)s(oth)g(cases.)47 b(Since)31 b(the)h(MPPs)g(of)g(graphs)3091 2210 y Fk(~)3077 2233 y(G)3149 2247 y Fj(t)3178 2233 y Fp(\()p Fk(X)3295 2200 y Fh(0)3335 2233 y Fp(\))382 2345 y(and)571 2322 y Fk(~)557 2345 y(G)629 2312 y Fv(\003)629 2368 y Fj(t)668 2345 y Fp(\()p Fk(X)785 2312 y Fh(0)825 2345 y Fp(\))e(are)f (di\013eren)m(t,)g(it)f(follo)m(ws)g(that)i(in)e(the)h(graph)2538 2322 y Fk(~)2521 2345 y(H)2597 2359 y Fj(t)2626 2345 y Fp(\()p Fk(X)2743 2312 y Fh(0)2784 2345 y Fp(\))g(there)g(are)g(t)m (w)m(o)382 2458 y(distinct)h(MPPs)i Fi(U)1053 2425 y Fv(0)1109 2458 y Fp(and)f Fi(U)1353 2425 y Fv(00)1395 2458 y Fp(,)i(whose)f(non-coinciden)m(t)f(arcs)i(form)e(a)i Fi(U)2883 2425 y Fv(0)2906 2458 y Fp(-alternating)382 2571 y(path)d Fk(P)43 b Fp(\(see)32 b(Theorem)e(7\),)h(con)m(taining)f (the)g(engaged)i(arc)f(\()p Fk(x)2584 2585 y Fj(i)2612 2571 y Fk(;)15 b(x)2704 2586 y Fj(k)r(;j)2799 2571 y Fp(\).)41 b(It)31 b(is)e(easy)i(to)382 2684 y(see)24 b(that)f(the)h(arc)f(\()p Fk(x)1092 2698 y Fj(i)1121 2684 y Fk(;)15 b(x)1213 2699 y Fj(k)r(;j)1308 2684 y Fp(\))23 b(b)s(elongs)f(to)i Fk(P)36 b Fp(if)22 b(and)g(only)g(if)g Fg(every)h Fp(head)g(of)h(an)f(engaged)382 2797 y(arc)37 b(of)f Fk(P)50 b Fp(is)35 b(inciden)m(t)g(to)j(a)f(free)f(arc)h(of)g (this)e(path,)j(and)e Fg(every)g Fp(tail)g(of)h(a)g(free)f(arc)382 2910 y(of)c Fk(P)45 b Fp(is)31 b(inciden)m(t)g(to)h(an)g(engaged)h(arc) g(of)f(this)f(path.)45 b(That)32 b(is,)f(the)i Fi(U)2883 2877 y Fv(0)2906 2910 y Fp(-alternating)382 3023 y(path)d Fk(P)43 b Fp(forms)30 b(an)g(ev)m(en)h(cycle)g Fk(C)1569 3038 y Fh(2)p Fj(k)1677 3023 y Fp(\()p Fk(k)e Fi(\025)c Fp(2\))31 b(con)m(taining)f(the)h(v)m(ertex)g Fk(x)2921 3037 y Fj(i;j)3001 3023 y Fp(.)523 3136 y(Replace)k(the)f(v)m(ertex)i Fk(x)1359 3150 y Fj(i;j)1473 3136 y Fp(b)m(y)e(the)h(arc)g(in)e(the)h Fi(P)2260 3103 y Fv(0)2284 3136 y Fp(-alternating)h(path)f Fk(P)13 b Fp(.)52 b(Ob)m(vi-)382 3249 y(ously)-8 b(,)30 b(w)m(e)h(obtain)f(an)g(o)s(dd)f(cycle)i Fk(C)1647 3264 y Fh(2)p Fj(k)r Fh(+1)1845 3249 y Fp(\()p Fk(k)e Fi(\025)c Fp(2\))31 b(of)g Fk(G)24 b Fi(2)h Fk(L)2511 3263 y Fj(n)2558 3249 y Fp(.)510 b(Q.E.D.)523 3461 y(Suc)m(h)30 b(an)g(o)s(dd)f(cycle)i (in)e(the)i(digraph)1883 3438 y Fk(~)1869 3461 y(G)p Fp(\()p Fk(X)2058 3428 y Fh(0)2098 3461 y Fp(\))g(can)f(b)s(e)g(rev)m (ealed)h(easily)-8 b(.)523 3574 y(In)m(terfere)28 b(paths)g(of)f(the)h (MPPs)g Fi(P)7 b Fp(,)29 b Fi(P)1852 3541 y Fv(\003)1920 3574 y Fp(and)e(delete)h(all)f(coinciding)f(arcs,)j(except)382 3687 y(the)k(arc)f Fk(x)746 3701 y Fj(i)803 3687 y Fk(>)d(x)955 3702 y Fj(k)r(;j)1049 3687 y Fp(.)47 b(Cho)s(osing)31 b(the)i(arc)g Fk(x)1885 3701 y Fj(i)1942 3687 y Fk(>)28 b(x)2093 3702 y Fj(k)r(;j)2220 3687 y Fp(as)33 b(initial)c(and)j(p)s (erforming)e(the)382 3811 y(searc)m(h)i(in)e(the)h(digraph)1271 3788 y Fk(~)1258 3811 y(G)o Fp(\()p Fk(X)1446 3778 y Fh(0)1486 3811 y Fp(\),)h(selecting)g(alternately)f(thic)m(k)g(and)g (thin)e(arcs)j(with)382 3924 y(opp)s(osite)e(directions,)f(w)m(e)i (will)c(\014nd)i(the)i(required)d(cycle.)523 4037 y(Th)m(us,)j(if)f (Case)i(2.2.2)i(o)s(ccurs)d(w)m(e)h(should)d(\014nd)h(the)i (restrictedly)e(stretc)m(hed)j(o)s(dd)382 4150 y(cycle)38 b(of)g Fk(G)p Fp(.)61 b(In)37 b(this)g(case)h(w)m(e)g(in)m(tro)s(duce)f (a)h(co-v)m(ertex,)k(that)c(is,)h(w)m(e)f(pass)f(to)h(the)382 4263 y(consideration)29 b(of)i(a)g(new)e(digraph.)382 4475 y Fq(Theorem)34 b(14)46 b Fg(L)-5 b(et)49 b Fi(P)57 b Fg(b)-5 b(e)49 b(the)g(MPP)f(of)1955 4452 y Fk(~)1941 4475 y(G)2013 4489 y Fj(t)2042 4475 y Fp(\()p Fk(X)2159 4442 y Fh(0)2200 4475 y Fp(\))p Fg(,)53 b(in)c(which)h(the)f(vertex)g Fk(x)3220 4489 y Fj(j)3306 4475 y Fg(is)382 4588 y(marke)-5 b(d.)73 b(Then)43 b(se)-5 b(quential)43 b(use)f(of)h(the)g(standar)-5 b(d)45 b(tr)-5 b(ansformation)47 b(with)c(r)-5 b(esp)g(e)g(ct)382 4701 y(to)31 b(the)g(marke)-5 b(d)31 b(vertex)g Fk(x)1271 4715 y Fj(j)1337 4701 y Fg(le)-5 b(ads)32 b(either)f(to)g(\014nding)g (an)f(MPP)f(of)2703 4678 y Fk(~)2689 4701 y(G)2761 4715 y Fj(t)2790 4701 y Fp(\()p Fk(X)2907 4668 y Fh(0)2947 4701 y Fp(\))i Fg(such)f(that)382 4814 y(the)j(vertex)g Fk(x)854 4828 y Fj(j)923 4814 y Fg(is)f(not)h(marke)-5 b(d,)34 b(or)g(to)f(the)g(c)-5 b(onstruction)34 b(of)f(a)g(new)g (VS-digr)-5 b(aph.)523 5026 y Fp(The)33 b(v)-5 b(alidit)m(y)32 b(of)i(the)f(theorem)h(follo)m(ws)f(from)g(the)g(analysis)f(of)i(the)g (considered)382 5139 y(ab)s(o)m(v)m(e)d(cases.)2202 b(Q.E.D.)1831 5652 y(31)p eop %%Page: 32 32 32 31 bop 382 548 a Fl(10)135 b(Classi\014cation)47 b(of)e(the)g(Mark)l (ed)g(V)-11 b(ertices)382 751 y Fp(F)j(or)34 b(further)f (consideration,)g(it)g(is)g(exp)s(edien)m(t)g(to)h(classify)f(the)g (mark)m(ed)h(v)m(ertices)g(of)382 864 y(a)d(transitiv)m(e)e(closure)h (graph.)523 977 y(Let)1322 1090 y Fk(H)7 b Fp(\()1454 1067 y Fk(~)1440 1090 y(G)1512 1104 y Fj(t)1541 1090 y Fp(\()p Fk(X)1658 1052 y Fh(0)1698 1090 y Fp(\)\))26 b(=)f Fi(fP)2005 1052 y Fh(1)2046 1090 y Fk(;)15 b(:)g(:)g(:)h(;)f Fi(P)2317 1052 y Fj(N)2386 1090 y Fi(g)382 1256 y Fp(b)s(e)30 b(the)g(set)h(of)g(all)e(MPPs)h(of)1424 1234 y Fk(~)1411 1256 y(G)1483 1270 y Fj(t)1512 1256 y Fp(\()p Fk(X)1629 1223 y Fh(0)1669 1256 y Fp(\).)523 1369 y(Designate:)1559 1555 y Fk(V)1612 1569 y Fh(1)1677 1555 y Fp(=)1807 1449 y Fj(N)1792 1474 y Ff(\\)1773 1658 y Fj(h)p Fh(=1)1919 1555 y Fi(B)s Fp(\()p Fi(P)2087 1517 y Fj(h)2133 1555 y Fp(\))p Fk(;)382 1817 y Fp(where)g Fi(P)715 1784 y Fj(h)786 1817 y Fi(2)24 b Fk(H)7 b Fp(\()1003 1794 y Fk(~)989 1817 y(G)1061 1831 y Fj(t)1091 1817 y Fp(\()p Fk(X)1208 1784 y Fh(0)1248 1817 y Fp(\)\).)523 1930 y(An)m(y)37 b(v)m(ertex)i(of)e(the)h(set)g Fk(V)1483 1944 y Fh(1)1559 1930 y Fp(is)f(called)f(an)h Fg(immovable)i Fp(in)d(the)h(graph)3065 1907 y Fk(~)3052 1930 y(G)3124 1944 y Fj(t)3153 1930 y Fp(\()p Fk(X)3270 1897 y Fh(0)3310 1930 y Fp(\).)382 2043 y(In)30 b(other)h(w)m(ords,)g(the)g(v)m(ertex)h Fk(x)1506 2057 y Fj(j)1568 2043 y Fi(2)26 b Fk(X)38 b Fp(is)30 b(immo)m(v)-5 b(able)30 b(in)f(the)i(graph)2850 2020 y Fk(~)2836 2043 y(G)2908 2057 y Fj(t)2937 2043 y Fp(\()p Fk(X)3054 2010 y Fh(0)3095 2043 y Fp(\))g(if)e(the)382 2156 y(v)m(ertex)i Fk(x)709 2170 y Fj(j)776 2156 y Fp(is)f(mark)m(ed)g (in)f(an)m(y)i(MPP)f(of)h(this)e(graph.)523 2278 y(The)24 b(MPP)h Fi(P)32 b Fp(of)1142 2255 y Fk(~)1129 2278 y(G)1201 2292 y Fj(t)1230 2278 y Fp(\()p Fk(X)1347 2245 y Fh(0)1387 2278 y Fp(\))25 b(is)f(called)f Fg(inclusion)28 b(maximal)e Fp(if)e(there)h(do)s(es)f(not)h(exist)382 2391 y(an)30 b(MPP)h Fi(P)816 2358 y Fv(\003)886 2391 y Fp(of)g(this)e(graph)h(suc)m (h)g(that)h Fi(B)s Fp(\()p Fi(P)7 b Fp(\))26 b Fi(\032)f(B)s Fp(\()p Fi(P)2320 2358 y Fv(\003)2360 2391 y Fp(\),)31 b Fi(P)i(6)p Fp(=)25 b Fi(P)7 b Fp(.)523 2504 y(Let)31 b Fk(H)769 2471 y Fv(\003)808 2504 y Fp(\()857 2481 y Fk(~)843 2504 y(G)915 2518 y Fj(t)945 2504 y Fp(\()p Fk(X)1062 2471 y Fh(0)1102 2504 y Fp(\)\))g(b)s(e)f(a)h(set)f(of)h(all) e(inclusion)f(maximal)h(MPPs)h(of)2915 2481 y Fk(~)2902 2504 y(G)2974 2518 y Fj(t)3003 2504 y Fp(\()p Fk(X)3120 2471 y Fh(0)3160 2504 y Fp(\).)523 2617 y(Designate:)1250 2730 y Fk(V)1303 2744 y Fh(2)1368 2730 y Fp(=)1703 2649 y Ff([)1464 2852 y Fv(8P)1559 2833 y Fb(h)1598 2852 y Fv(2)p Fj(H)1708 2833 y Fo(\003)1744 2852 y Fh(\()1782 2835 y Fj(~)1771 2852 y(G)1826 2860 y Fb(t)1855 2852 y Fh(\()p Fj(X)1945 2833 y Fe(0)1980 2852 y Fh(\)\))2050 2730 y Fi(B)s Fp(\()p Fi(P)2218 2693 y Fj(h)2263 2730 y Fp(\))21 b Fi(n)g Fk(V)2438 2744 y Fh(1)2477 2730 y Fk(:)523 3027 y Fp(An)m(y)32 b(v)m(ertex)h(of)f(the)g(set)h Fk(V)1456 3041 y Fh(2)1527 3027 y Fp(is)e(called)g Fg(pseudo-movable)j Fp(in)c(the)i(graph)3065 3004 y Fk(~)3052 3027 y(G)3124 3041 y Fj(t)3153 3027 y Fp(\()p Fk(X)3270 2994 y Fh(0)3310 3027 y Fp(\).)382 3140 y(In)39 b(other)i(w)m(ords,)h(the)e(v)m(ertex)i Fk(x)1555 3154 y Fj(j)1633 3140 y Fi(2)f Fk(X)48 b Fp(is)39 b(pseudo-mo)m(v)-5 b(able)40 b(in)2755 3117 y Fk(~)2741 3140 y(G)2813 3154 y Fj(t)2843 3140 y Fp(\()p Fk(X)2960 3107 y Fh(0)3000 3140 y Fp(\))g(if)f(there)382 3253 y(exist)29 b(the)h(inclusion)d(maximal)i(MPPs)h Fi(P)37 b Fp(and)29 b Fi(P)2121 3220 y Fv(\003)2191 3253 y Fp(of)h(this)f(graph)g(suc)m(h)g (that)i(in)d(one)382 3366 y(of)i(them)h(the)f(v)m(ertex)i Fk(x)1202 3380 y Fj(j)1269 3366 y Fp(is)d(mark)m(ed,)i(and)f(in)f(the)h (other)h(one)f(is)g(unmark)m(ed.)523 3489 y(A)39 b(mark)m(ed)g(v)m (ertex)i Fk(x)1299 3503 y Fj(j)1375 3489 y Fi(2)e(B)s Fp(\()p Fi(P)7 b Fp(\))40 b(of)1844 3466 y Fk(~)1831 3489 y(G)1903 3503 y Fj(t)1932 3489 y Fp(\()p Fk(X)2049 3456 y Fh(0)2089 3489 y Fp(\))f(b)s(eing)f(suc)m(h)h(that)h Fk(x)2889 3503 y Fj(j)2965 3489 y Fi(62)f Fk(V)3118 3503 y Fh(1)3158 3489 y Fp(,)i(and)382 3611 y Fk(x)434 3625 y Fj(j)496 3611 y Fi(62)24 b Fk(V)634 3625 y Fh(2)674 3611 y Fp(,)31 b(is)e(called)g(a)i Fg(movable)h Fp(v)m(ertex.)41 b(Clearly)-8 b(,)30 b Fi(P)j(62)25 b Fk(H)2410 3578 y Fv(\003)2449 3611 y Fp(\()2498 3588 y Fk(~)2484 3611 y(G)2556 3625 y Fj(t)2586 3611 y Fp(\()p Fk(X)2703 3578 y Fh(0)2743 3611 y Fp(\)\))31 b(in)e(this)g(case.)p 869 4295 1 473 v 869 3906 a Fc(A)911 3989 y(A)952 4072 y(A)994 4155 y(A)1035 4238 y(A)1064 4295 y(A)o(\001)1147 4212 y(\001)1188 4129 y(\001)1230 4046 y(\001)1271 3963 y(\001)1300 3906 y(\001)1109 4298 y(\001)1150 4215 y(\001)1192 4132 y(\001)1233 4049 y(\001)1275 3966 y(\001)1303 3909 y(\001)1112 4302 y(\001)1154 4219 y(\001)1195 4136 y(\001)1237 4053 y(\001)1278 3970 y(\001)1307 3912 y(\001)1102 4292 y(\001)1144 4209 y(\001)1185 4126 y(\001)1227 4043 y(\001)1268 3960 y(\001)1297 3902 y(\001)1099 4288 y(\001)1140 4205 y(\001)1182 4122 y(\001)1223 4039 y(\001)1265 3956 y(\001)1293 3899 y(\001)p 1341 4295 V 1342 4295 a Fd(x)-313 b(x)j(x)869 4059 y(x)398 b(x)869 3823 y(x)g(x)823 4414 y Fk(x)875 4428 y Fh(1)1058 4414 y Fk(x)1110 4428 y Fh(2)1296 4414 y Fk(x)1348 4428 y Fh(3)729 4040 y Fk(x)781 4054 y Fh(4)1402 4037 y Fk(x)1454 4051 y Fh(5)823 3748 y Fk(x)875 3762 y Fh(6)1301 3748 y Fk(x)1353 3762 y Fh(7)1047 4530 y Fp(\(a\))p 1696 4295 V 1696 3906 a Fc(A)1737 3989 y(A)1779 4072 y(A)1820 4155 y(A)1862 4238 y(A)1891 4295 y(A)1699 3902 y(A)1741 3985 y(A)1782 4068 y(A)1824 4151 y(A)1865 4234 y(A)1894 4292 y(A)1703 3899 y(A)1744 3982 y(A)1786 4065 y(A)1827 4148 y(A)1869 4231 y(A)1897 4288 y(A)1693 3909 y(A)1734 3992 y(A)1776 4075 y(A)1817 4158 y(A)1859 4241 y(A)1887 4298 y(A)1689 3912 y(A)1731 3995 y(A)1772 4078 y(A)1814 4161 y(A)1855 4244 y(A)1884 4302 y(A)1932 4295 y(\001)1974 4212 y(\001)2015 4129 y(\001)2057 4046 y(\001)2098 3963 y(\001)2127 3906 y(\001)p 2168 4295 V 2168 4295 a Fd(x)-314 b(x)-308 b(x)1696 4059 y(x)397 b(x)1696 3823 y(x)g(x)1650 4414 y Fk(x)1702 4428 y Fh(1)1883 4414 y Fk(x)1935 4428 y Fh(2)2123 4414 y Fk(x)2175 4428 y Fh(3)1556 4040 y Fk(x)1608 4054 y Fh(4)2217 4037 y Fk(x)2269 4051 y Fh(5)1650 3748 y Fk(x)1702 3762 y Fh(6)2123 3748 y Fk(x)2175 3762 y Fh(7)1872 4530 y Fp(\(b\))p 2522 4295 V 2523 3906 a Fc(A)2564 3989 y(A)2606 4072 y(A)2647 4155 y(A)2689 4238 y(A)2717 4295 y(A)p 2759 4295 V -389 x(A)2800 3989 y(A)2842 4072 y(A)2883 4155 y(A)2925 4238 y(A)2954 4295 y(A)2762 3902 y(A)2804 3985 y(A)2845 4068 y(A)2887 4151 y(A)2928 4234 y(A)2957 4292 y(A)2766 3899 y(A)2807 3982 y(A)2849 4065 y(A)2890 4148 y(A)2932 4231 y(A)2960 4288 y(A)2756 3909 y(A)2797 3992 y(A)2839 4075 y(A)2880 4158 y(A)2922 4241 y(A)2950 4298 y(A)2752 3912 y(A)2794 3995 y(A)2835 4078 y(A)2877 4161 y(A)2918 4244 y(A)2947 4302 y(A)2523 4295 y Fd(x)161 b(x)g(x)2523 4059 y(x)g(x)2523 3823 y(x)g(x)2477 4414 y Fk(x)2529 4428 y Fh(1)2713 4414 y Fk(x)2765 4428 y Fh(2)2949 4414 y Fk(x)3001 4428 y Fh(3)2380 4030 y Fk(x)2432 4044 y Fh(4)2611 3961 y Fk(x)2663 3975 y Fh(5)2477 3748 y Fk(x)2529 3762 y Fh(6)2713 3748 y Fk(x)2765 3762 y Fh(7)2703 4530 y Fp(\(c\))p 857 4295 25 473 v 1329 4295 25 237 v 1683 4295 V 2156 4295 25 473 v 2510 4295 V 2746 4295 25 237 v 837 4727 a(Fig.)31 b(14:)41 b(Distinct)30 b(MPPs)g(of)h(a)f (transitiv)m(e)g(closure)g(graph)523 4931 y(Fig.)69 b(2)40 b(\(a\))g(sho)m(ws)g(the)g(mo)m(v)-5 b(able)39 b(v)m(ertices)i Fk(x)2177 4945 y Fh(5)2216 4931 y Fp(,)h Fk(x)2335 4945 y Fh(6)2375 4931 y Fp(,)g(Fig.)68 b(14)41 b(\(a\),)i(\(b\))d(sho)m(ws) 382 5044 y(the)34 b(pseudo-mo)m(v)-5 b(able)33 b(v)m(ertices)h Fk(x)1586 5058 y Fh(6)1625 5044 y Fp(,)h Fk(x)1737 5058 y Fh(7)1809 5044 y Fp(resp)s(ectiv)m(ely)-8 b(,)35 b(and)d(Fig.)50 b(14)35 b(\(c\))f(sho)m(ws)f(the)382 5157 y(immo)m(v)-5 b(able)29 b(v)m(ertex)j Fk(x)1164 5171 y Fh(6)1203 5157 y Fp(.)1831 5652 y(32)p eop %%Page: 33 33 33 32 bop 382 548 a Fl(11)135 b(The)44 b(Algorithm)i(for)f(the)g (Solution)h(of)f(MCPP)382 751 y Fp(In)27 b(Section)h(9)g(of)g(this)f (pap)s(er,)h(w)m(e)g(in)m(tro)s(duced)e(the)i(concept)h(of)f(the)h (standard)e(trans-)382 864 y(formation)34 b(and)g(ha)m(v)m(e)i (examined)d(the)i(results)e(of)i(suc)m(h)f(a)h(transformation.)53 b(In)33 b(the)382 977 y(presen)m(t)27 b(section,)g(w)m(e)g(con)m(tin)m (ue)g(this)f(researc)m(h)h(using)e(the)i(in)m(tro)s(duced)e (classi\014cation)382 1090 y(of)31 b(the)f(mark)m(ed)h(v)m(ertices.)42 b(T)-8 b(ogether)31 b(with)f(the)g(construction)h(of)f(the)h(analyzing) f(di-)382 1203 y(graph,)g(this)f(transformation)h(allo)m(ws)g(us)f(to)j (form)m(ulate)e(an)g(algorithm)g(to)h(solv)m(e)f(the)382 1315 y(Minim)m(um)e(Clique)g(P)m(artition)i(Problem)f(for)i(a)f(graph)g Fk(G)25 b Fi(2)g Fk(L)2552 1329 y Fj(n)2599 1315 y Fp(.)382 1528 y Fq(Claim)33 b(4)46 b Fg(L)-5 b(et)42 b(ther)-5 b(e)44 b(b)-5 b(e)42 b(a)g(digr)-5 b(aph)1743 1505 y Fk(~)1729 1528 y(G)p Fp(\()p Fk(X)1918 1495 y Fh(0)1958 1528 y Fp(\))43 b Fg(and)g(let)f(some)h(MPP)e Fi(P)50 b Fg(of)3091 1505 y Fk(~)3077 1528 y(G)3149 1542 y Fj(t)3178 1528 y Fp(\()p Fk(X)3295 1495 y Fh(0)3335 1528 y Fp(\))382 1641 y Fg(b)-5 b(e)41 b(found,)i(in)d(which)i(a)f(vertex)g Fk(x)1592 1655 y Fj(j)1669 1641 y Fg(is)g(movable.)67 b(Then)41 b(a)g(\014nite)g(se)-5 b(quenc)g(e)40 b(of)h(the)382 1764 y(standar)-5 b(d)40 b(tr)-5 b(ansformation)41 b(of)c(the)h(digr)-5 b(aph)1988 1741 y Fk(~)1975 1764 y(G)o Fp(\()p Fk(X)2163 1731 y Fh(0)2204 1764 y Fp(\))37 b Fg(le)-5 b(ads)38 b(to)g(the)g(c)-5 b(onstruction)38 b(of)382 1877 y(a)33 b(new)g(MPP)e Fi(P)954 1844 y Fv(0)1011 1877 y Fg(of)1131 1854 y Fk(~)1118 1877 y(G)1190 1891 y Fj(t)1219 1877 y Fp(\()p Fk(X)1336 1844 y Fh(0)1376 1877 y Fp(\))p Fg(.)523 2089 y Fp(Let)h(the)g(conditions)e(of)i(Theorem)f(4)h(b)s(e)f (satis\014ed.)44 b(By)32 b(the)g(de\014nition)d(of)j(mo)m(v-)382 2202 y(able)k(v)m(ertex)i Fk(x)914 2216 y Fj(j)987 2202 y Fi(2)d Fk(X)7 b Fp(,)39 b(the)e(TCG)1645 2179 y Fk(~)1631 2202 y(G)1703 2216 y Fj(t)1732 2202 y Fp(\()p Fk(X)1849 2169 y Fh(0)1890 2202 y Fp(\))g(has)f(an)h(MPP)g Fi(P)2577 2169 y Fv(0)2637 2202 y Fp(in)f(whic)m(h)f Fk(x)3068 2216 y Fj(j)3142 2202 y Fp(is)g(not)382 2325 y(mark)m(ed.)57 b(Therefore,)37 b(the)f(TCG)1615 2302 y Fk(~)1602 2325 y(G)1674 2292 y Fv(0)1674 2347 y Fj(t)1703 2325 y Fp(\()p Fk(X)1820 2292 y Fh(0)1860 2325 y Fp(\))g(of)g(an)g(auxiliary)d (digraph)2909 2302 y Fk(~)2896 2325 y(G)2968 2292 y Fv(0)2990 2325 y Fp(\()p Fk(X)3107 2292 y Fh(0)3148 2325 y Fp(\))j(also)382 2438 y(has)e(the)g(MPP)g Fi(P)1019 2405 y Fv(0)1043 2438 y Fp(.)51 b(Therefore,)35 b(the)f(execution)g(of)g(the)g(standard)f (transformation)382 2551 y(alw)m(a)m(ys)23 b(leads)e(to)i(Case)g(1.)38 b(And,)23 b(if)e(Case)i(1.1)g(o)s(ccurs)f(\(the)g(v)m(ertex)i Fk(x)2722 2565 y Fj(j)2780 2551 y Fp(is)e(not)g(mark)m(ed\),)382 2664 y(then)30 b(the)h(construction)f(of)h(the)g(required)e(sequence)i (of)g(standard)e(transformations)382 2776 y(is)37 b(completed.)64 b(If)37 b(Case)i(1.2)g(tak)m(es)g(place,)h(then,)g(in)d(a)h(new)g (digraph,)g(the)g(v)m(ertex)382 2889 y Fk(x)434 2903 y Fj(j)470 2889 y Fp(,)28 b(ob)m(viously)-8 b(,)26 b(remains)g(mo)m(v) -5 b(able)26 b(and)g(the)h(new)f(standard)f(transformation)h(is)f(p)s (er-)382 3012 y(formed.)42 b(It)32 b(is)e(clear)h(that)h(b)s(ecause)f (of)h(the)f(\014niteness)f(of)h(the)g(digraph)2937 2989 y Fk(~)2924 3012 y(G)o Fp(\()p Fk(X)3112 2979 y Fh(0)3153 3012 y Fp(\),)g(the)382 3125 y(n)m(um)m(b)s(er)e(of)h(suc)m(h)h (transformations)e(is)h(also)g(\014nite.)914 b(Q.E.D.)523 3338 y(Let)38 b(us)e(consider)g(an)h(application)e(of)i(the)g(standard) g(transformation)f(of)h(a)h(di-)382 3450 y(graph,)30 b(when)f(the)i(mark)m(ed)f(v)m(ertex)i Fk(x)1710 3464 y Fj(j)1777 3450 y Fp(is)d(pseudo-mo)m(v)-5 b(able.)523 3573 y(Generally)d(,)25 b(in)d(the)i(TCG)1437 3550 y Fk(~)1424 3573 y(G)1496 3587 y Fj(t)1525 3573 y Fp(\()p Fk(X)1642 3540 y Fh(0)1682 3573 y Fp(\),)i(there)e(are)g(sev)m(eral)g (di\013eren)m(t)f(MPPs)h(con)m(tain-)382 3686 y(ing)34 b(the)h(pseudo-mo)m(v)-5 b(able)35 b(v)m(ertex)h(as)f(mark)m(ed)g(with) e(di\013eren)m(t)i(leading)f(arcs.)54 b(The)382 3799 y(set)31 b(of)f(all)g(suc)m(h)g(leading)f(arcs)h(w)m(e)h(designate)g (as)f Fk(M)10 b Fp(\()p Fk(x)2289 3813 y Fj(j)2326 3799 y Fp(.)523 3912 y(Let)32 b Fk(x)739 3926 y Fj(j)807 3912 y Fp(b)s(e)f(a)g(pseudo-mo)m(v)-5 b(able)31 b(v)m(ertex)i(in)d(an)h (MPP)h Fi(P)7 b Fp(.)44 b(A)32 b(v)m(ertex)38 b(~)-51 b Fk(x)2983 3926 y Fj(j)3051 3912 y Fp(is)30 b(called)382 4025 y(a)40 b Fg(p)-5 b(artner)42 b Fp(of)e(the)h(pseudo-mo)m(v)-5 b(able)39 b(v)m(ertex)j Fk(x)2079 4039 y Fj(j)2155 4025 y Fp(if)j(~)-51 b Fk(x)2300 4039 y Fj(j)2376 4025 y Fp(is)39 b(mark)m(ed)i(in)d(some)j(MPP)382 4148 y(of)502 4125 y Fk(~)489 4148 y(G)561 4162 y Fj(t)590 4148 y Fp(\()p Fk(X)707 4115 y Fh(0)747 4148 y Fp(\))34 b(i\013)e(an)m(y)i(of)g(arcs)f (of)h(the)g(set)g Fk(M)10 b Fp(\()p Fk(x)1996 4162 y Fj(j)2032 4148 y Fp(\))34 b(is)f(not)g(leading)g(for)g(the)g(v)m(ertex) i Fk(x)3309 4162 y Fj(j)3345 4148 y Fp(.)382 4261 y(Ob)m(viously)-8 b(,)42 b(the)f(concept)h(of)g(the)f(partnership)d(is)i(comm)m(utativ)m (e,)46 b(that)41 b(is,)i(if)d(the)382 4374 y(v)m(ertex)48 b(~)-51 b Fk(x)720 4388 y Fj(j)797 4374 y Fp(is)40 b(a)h(partner)f(of)h (the)g(v)m(ertex)h Fk(x)1938 4388 y Fj(j)1975 4374 y Fp(,)i(then)c(and)g(the)h(v)m(ertex)h Fk(x)2953 4388 y Fj(j)3031 4374 y Fp(is)d(also)i(a)382 4486 y(partner)30 b(of)g(the)h(v)m(ertex)37 b(~)-51 b Fk(x)1293 4500 y Fj(j)1330 4486 y Fp(.)523 4609 y(During)22 b(of)i(execution)g(of)g(the) g(standard)f(transformation)g(of)2663 4586 y Fk(~)2650 4609 y(G)o Fp(\()p Fk(X)2838 4576 y Fh(0)2878 4609 y Fp(\),)j(it)d(is)g(neces-)382 4722 y(sary)k(to)g(distinguish)c(t)m(w)m (o)29 b(t)m(yp)s(es)e(of)g(pseudo-mo)m(v)-5 b(able)26 b(v)m(ertices)i(of)2737 4699 y Fk(~)2723 4722 y(G)2795 4736 y Fj(t)2824 4722 y Fp(\()p Fk(X)2941 4689 y Fh(0)2981 4722 y Fp(\).)40 b(W)-8 b(e)28 b(call)382 4835 y(a)34 b(pseudo-mo)m(v)-5 b(able)33 b(v)m(ertex)i Fk(x)1452 4849 y Fj(j)1522 4835 y Fg(fr)-5 b(e)g(e)p Fp(,)35 b(if)e(at)h(least)g (one)g(of)g(arcs)g(of)g(the)g(set)g Fk(M)10 b Fp(\()c(~)-51 b Fk(x)3204 4849 y Fj(j)3241 4835 y Fp(\))34 b(is)382 4948 y(not)28 b(mark)m(ed)h(as)f(\\considered,")h(where)k(~)-51 b Fk(x)1844 4962 y Fj(j)1909 4948 y Fp(is)27 b(a)h(partner)g(of)g Fk(x)2546 4962 y Fj(j)2583 4948 y Fp(.)40 b(A)28 b(pseudo-mo)m(v)-5 b(able)382 5061 y(v)m(ertex)45 b Fk(x)723 5075 y Fj(j)804 5061 y Fp(is)e(called)g Fg(c)-5 b(onne)g(cte)g(d)46 b Fp(if)d(all)g(arcs)i(of)f(the)g(set)h Fk(M)10 b Fp(\()c(~)-51 b Fk(x)2670 5075 y Fj(j)2707 5061 y Fp(\))45 b(are)f(mark)m(ed)g(as)382 5174 y(\\considered.")523 5287 y(Ob)m(viously)-8 b(,)25 b(b)m(y)g(the)g(de\014nition)e(of)i(the)h(standard)e(transformation)g (of)i(a)f(digraph,)382 5400 y(a)37 b(connected)h(pseudo-mo)m(v)-5 b(able)36 b(v)m(ertex)i(should)d(b)s(e)h(considered)f(as)i(immo)m(v)-5 b(able)36 b(in)1831 5652 y(33)p eop %%Page: 34 34 34 33 bop 382 548 a Fp(the)30 b(TCG)785 525 y Fk(~)771 548 y(G)843 562 y Fj(t)873 548 y Fp(\()p Fk(X)990 515 y Fh(0)1030 548 y Fp(\).)382 738 y Fq(Claim)j(5)46 b Fg(L)-5 b(et)32 b Fk(x)994 752 y Fj(j)1063 738 y Fg(b)-5 b(e)32 b(a)h(fr)-5 b(e)g(e)32 b(pseudo-movable)i(vertex)e(in)g(the)h (MPP)e Fi(P)40 b Fg(of)3063 715 y Fk(~)3049 738 y(G)3121 752 y Fj(t)3150 738 y Fp(\()p Fk(X)3267 705 y Fh(0)3307 738 y Fp(\))p Fg(.)382 851 y(Then)34 b(the)h(one-p)-5 b(ass)35 b(exe)-5 b(cution)35 b(of)f(the)h(standar)-5 b(d)37 b(tr)-5 b(ansformation)37 b(with)e(r)-5 b(esp)g(e)g(ct)36 b(to)382 964 y(the)k(marke)-5 b(d)40 b(vertex)g Fk(x)1191 978 y Fj(j)1266 964 y Fg(le)-5 b(ads)41 b(to)f(the)f(c)-5 b(onstruction)41 b(of)f(MPP)e Fi(P)2723 931 y Fv(0)2786 964 y Fg(of)h(the)h(TCG)f(of)382 1087 y(the)e(auxiliary)h(digr)-5 b(aph,)40 b(which)e(is)f(also)h(an)g(MPP)d(of)2327 1064 y Fk(~)2314 1087 y(G)2386 1101 y Fj(t)2415 1087 y Fp(\()p Fk(X)2532 1054 y Fh(0)2572 1087 y Fp(\))p Fg(,)j(and)g(the)f(vertex)g Fk(x)3334 1101 y Fj(j)382 1200 y Fg(satis\014es)c(one)g(of)g(the)g(fol) 5 b(lowing)34 b(c)-5 b(onditions:)518 1369 y Fi(\017)46 b Fg(the)33 b(vertex)g Fk(x)1081 1383 y Fj(j)1150 1369 y Fg(is)f(not)i(marke)-5 b(d)34 b(in)e Fi(P)1906 1336 y Fv(0)1930 1369 y Fg(;)518 1549 y Fi(\017)46 b Fg(the)33 b(marke)-5 b(d)34 b(vertex)f Fk(x)1398 1563 y Fj(j)1467 1549 y Fg(is)g(a)g(fr)-5 b(e)g(e)33 b(pseudo-movable)h(vertex;)518 1729 y Fi(\017)46 b Fg(the)33 b(marke)-5 b(d)34 b(vertex)f Fk(x)1398 1743 y Fj(j)1467 1729 y Fg(is)g(a)g(c)-5 b(onne)g(cte)g(d)34 b(pseudo-movable)g(vertex.)523 1930 y Fp(Indeed,)26 b(let)h Fk(x)1020 1944 y Fj(j)1082 1930 y Fp(b)s(e)f(a)g(free)h(pseudo-mo)m(v) -5 b(able)26 b(v)m(ertex)h(in)e(the)h(MPP)h Fi(P)33 b Fp(of)3065 1907 y Fk(~)3052 1930 y(G)3124 1944 y Fj(t)3153 1930 y Fp(\()p Fk(X)3270 1897 y Fh(0)3310 1930 y Fp(\).)382 2043 y(Execute)d(the)f(standard)f(transformation)h(with)e(resp)s(ect)j (to)f(the)h(mark)m(ed)f(v)m(ertex)h Fk(x)3309 2057 y Fj(j)3345 2043 y Fp(.)382 2155 y(Then,)40 b(b)m(y)f(the)h(de\014nition) c(of)j(a)h(free)f(pseudo-mo)m(v)-5 b(able)39 b(v)m(ertex,)j(the)e(TCG)e (of)h(the)382 2268 y(auxiliary)28 b(digraph)h(has)h(an)g(MPP)g Fi(P)1690 2235 y Fv(0)1745 2268 y Fp(that)h(is)e(also)h(an)h(MPP)f(of) 2696 2245 y Fk(~)2682 2268 y(G)2754 2282 y Fj(t)2784 2268 y Fp(\()p Fk(X)2901 2235 y Fh(0)2941 2268 y Fp(\).)523 2381 y(Ob)m(viously)-8 b(,)32 b(if)f(arcs)i(of)g(the)g(set)g Fk(M)10 b Fp(\()p Fk(x)1840 2395 y Fj(j)1877 2381 y Fp(\))33 b(do)f(not)h(b)s(elong)f(to)h(the)g(MPP)g Fi(P)3112 2348 y Fv(0)3136 2381 y Fp(,)g(then)382 2494 y(the)26 b(v)m(ertex)g Fk(x)856 2508 y Fj(j)918 2494 y Fp(is)f(not)g(mark)m(ed)h(in)e(this)g (MPP)-8 b(.)26 b(Otherwise,)g(the)g(v)m(ertex)g Fk(x)2929 2508 y Fj(j)2991 2494 y Fp(is)f(mark)m(ed)382 2607 y(in)k Fi(P)558 2574 y Fv(0)582 2607 y Fp(.)523 2720 y(Let)43 b(~)-51 b Fk(x)744 2734 y Fj(j)817 2720 y Fp(b)s(e)35 b(a)i(partner)f(of)g(the)h(v)m(ertex)h Fk(x)1964 2734 y Fj(j)2036 2720 y Fp(in)2162 2697 y Fk(~)2148 2720 y(G)2220 2734 y Fj(t)2249 2720 y Fp(\()p Fk(X)2366 2687 y Fh(0)2407 2720 y Fp(\))e(and)g(all)f(arcs)i(of)g(the)f(set)382 2833 y Fk(M)10 b Fp(\()c(~)-51 b Fk(x)567 2847 y Fj(j)604 2833 y Fp(\))37 b(b)s(e)f(mark)m(ed)h(as)f(\\considered.")h(Then,)h(ob) m(viously)-8 b(,)37 b(an)g(MPP)f(of)h(the)g(TCG)382 2946 y(of)h(the)g(auxiliary)e(digraph,)i(whic)m(h)f(is)g(also)h(an)f(MPP)h (of)g(the)h(TCG)e(of)h(the)g(initial)382 3059 y(graph)29 b(ha)m(ving)g(the)h(v)m(ertex)h Fk(x)1414 3073 y Fj(j)1480 3059 y Fp(as)f(not)g(mark)m(ed,)h(cannot)f(b)s(e)f(obtained)g(b)m(y)h (means)f(of)382 3172 y(next)g(standard)g(transformations.)40 b(In)28 b(other)i(w)m(ords,)f(in)f(this)g(case)j(the)e(v)m(ertex)i Fk(x)3244 3186 y Fj(j)3310 3172 y Fp(is)382 3294 y(a)g(connected)g (pseudo-mo)m(v)-5 b(able)30 b(v)m(ertex)h(of)g(the)g(TCG)2321 3271 y Fk(~)2308 3294 y(G)2380 3308 y Fj(t)2409 3294 y Fp(\()p Fk(X)2526 3261 y Fh(0)2566 3294 y Fp(\).)523 3407 y(Clearly)-8 b(,)27 b(if)g(at)h(least)g(one)f(arc)h(of)g(the)f (set)h(is)f(not)h(mark)m(ed)f(as)h(\\considered",)g(then)382 3520 y(the)i(v)m(ertex)i Fk(x)866 3534 y Fj(j)933 3520 y Fp(is)d(a)i(free)f(pseudo-mo)m(v)-5 b(able)30 b(v)m(ertex.)893 b(Q.E.D.)382 3720 y Fq(Claim)33 b(6)46 b Fg(L)-5 b(et)33 b(ther)-5 b(e)34 b(b)-5 b(e)33 b(a)g(digr)-5 b(aph)1696 3697 y Fk(~)1683 3720 y(G)o Fp(\()p Fk(X)1871 3687 y Fh(0)1912 3720 y Fp(\))33 b Fg(and)h(let)f(some)g(MPP)f Fi(P)40 b Fg(of)2979 3697 y Fk(~)2965 3720 y(G)3037 3734 y Fj(t)3066 3720 y Fp(\()p Fk(X)3183 3687 y Fh(0)3224 3720 y Fp(\))33 b Fg(b)-5 b(e)382 3833 y(found,)43 b(in)d(which)h(the)g (vertex)g Fk(x)1543 3847 y Fj(j)1619 3833 y Fg(is)g(immovable)g(or)g(c) -5 b(onne)g(cte)g(d)42 b(pseudo-movable.)382 3946 y(Then)29 b(a)g(one-p)-5 b(ass)30 b(exe)-5 b(cution)29 b(of)g(the)g(standar)-5 b(d)31 b(tr)-5 b(ansformation)33 b(le)-5 b(ads)30 b(to)f(obtaining)382 4059 y(an)k(MPP)e Fi(P)819 4026 y Fv(0)876 4059 y Fg(of)i(the)g(TCG)g (of)g(the)g(auxiliary)g(digr)-5 b(aph)2338 4036 y Fk(~)2324 4059 y(G)2396 4026 y Fv(0)2419 4059 y Fp(\()p Fk(X)2536 4026 y Fh(0)2576 4059 y Fp(\))33 b Fg(such)g(that)1440 4241 y Fk(C)7 b(ar)s(d)p Fp(\()p Fi(P)1756 4204 y Fv(0)1779 4241 y Fp(\))26 b Fk(>)f(C)7 b(ar)s(d)p Fp(\()p Fi(P)g Fp(\))p Fk(:)523 4432 y Fp(Let)28 b(the)g(conditions)e(of)i(the)g (claim)f(6)h(b)s(e)f(satis\014ed.)39 b(Since)27 b(the)h(initiating)d (sets)j(of)395 4521 y Fk(~)382 4544 y(G)o Fp(\()p Fk(X)570 4511 y Fh(0)611 4544 y Fp(\))i(and)867 4521 y Fk(~)853 4544 y(G)925 4511 y Fv(0)948 4544 y Fp(\()p Fk(X)1065 4511 y Fh(0)1105 4544 y Fp(\))h(are)f(the)h(same,)g(then)1440 4727 y Fk(C)7 b(ar)s(d)p Fp(\()p Fi(P)1756 4689 y Fv(0)1779 4727 y Fp(\))26 b Fi(\025)f Fk(C)7 b(ar)s(d)p Fp(\()p Fi(P)g Fp(\))p Fk(:)523 4909 y Fp(Supp)s(ose)29 b(that)1440 5021 y Fk(C)7 b(ar)s(d)p Fp(\()p Fi(P)1756 4984 y Fv(0)1779 5021 y Fp(\))26 b(=)f Fk(C)7 b(ar)s(d)p Fp(\()p Fi(P)g Fp(\))p Fk(:)382 5177 y Fp(Since)36 b(all)h(arcs)g(of)h(the)g (auxiliary)d(digraph)g(are)j(also)g(arcs)f(of)h(the)f(initial)e (digraph,)382 5290 y(the)i(MPP)h Fi(P)860 5257 y Fv(0)921 5290 y Fp(is)e(also)h(an)h(MPP)f(of)g(the)h(iniatial)d(digraph.)60 b(W)-8 b(e)38 b(ha)m(v)m(e)h(obtained)d(a)382 5403 y(con)m(tradiction) 30 b(with)f(the)i(de\014nition)d(of)i(an)h(immo)m(v)-5 b(able)29 b(v)m(ertex.)437 b(Q.E.D.)1831 5652 y(34)p eop %%Page: 35 35 35 34 bop 523 548 a Fp(So,)27 b(let)e(there)h(b)s(e)f(a)h(graph)f Fk(G)g Fi(2)g Fk(L)1713 562 y Fj(n)1760 548 y Fp(.)39 b(T)-8 b(o)26 b(solv)m(e)g(the)g(Minim)m(um)d(Clique)h(P)m(artition)382 661 y(Problem)29 b(for)h Fk(G)p Fp(,)g(it)g(is)g(su\016cien)m(t)f(to)i (execute)h(the)f(follo)m(wing)e(steps.)609 846 y Fq(The)35 b(algorithm)f(CLIP)-9 b(A)g(T-M.)382 982 y Fg(Step)33 b(1.)46 b Fp(Construct)30 b(some)h(aciclic)e(digraph)1977 959 y Fk(~)1963 982 y(G)o Fp(\()p Fk(X)2151 949 y Fh(0)2192 982 y Fp(\).)382 1119 y Fg(Step)k(2.)46 b Fp(Construct)30 b(a)h(VS-digraph)1701 1096 y Fk(~)1687 1119 y(G)p Fp(\()p Fk(X)1876 1086 y Fh(0)1916 1119 y Fp(\).)382 1255 y Fg(Step)i(3.)46 b Fp(Construct)30 b(the)g(transitiv)m(e)g(closure)g(graph)2265 1232 y Fk(~)2252 1255 y(G)2324 1269 y Fj(t)2353 1255 y Fp(\()p Fk(X)2470 1222 y Fh(0)2510 1255 y Fp(\).)382 1392 y Fg(Step)j(4.)46 b Fp(Find)29 b(an)h(MPP)g Fi(P)38 b Fp(of)1502 1369 y Fk(~)1488 1392 y(G)1560 1406 y Fj(t)1589 1392 y Fp(\()p Fk(X)1706 1359 y Fh(0)1746 1392 y Fp(\).)382 1528 y Fg(Step)33 b(5.)46 b Fp(In)28 b(the)i(TCG)1217 1505 y Fk(~)1204 1528 y(G)1276 1542 y Fj(t)1305 1528 y Fp(\()p Fk(X)1422 1495 y Fh(0)1462 1528 y Fp(\),)g(\014nd)e(a)h(mark) m(ed)h(v)m(ertex)g Fk(x)2456 1542 y Fj(j)2493 1528 y Fp(,)f(ha)m(ving)g(the)g(minim)m(um)609 1641 y(rank.)40 b(If)30 b(suc)m(h)g(a)h(v)m(ertex)h(do)s(es)e(not)g(exist,)h(then)f(go) h(to)g(Step)f(13.)382 1778 y Fg(Step)j(6.)46 b Fp(Execute)32 b(the)g(standard)f(transformation)f(of)2330 1755 y Fk(~)2316 1778 y(G)o Fp(\()p Fk(X)2504 1745 y Fh(0)2545 1778 y Fp(\))i(with)e(resp)s(ect)h(to)h(the)609 1891 y(mark)m(ed)f(v)m(ertex)g Fk(x)1260 1905 y Fj(j)1297 1891 y Fp(.)382 2027 y Fg(Step)i(7.)46 b Fp(If)35 b Fk(C)7 b(ar)s(d)p Fp(\()p Fi(P)1116 1994 y Fv(0)1140 2027 y Fp(\))34 b Fk(>)g(C)7 b(ar)s(d)p Fp(\()p Fi(P)g Fp(\),)38 b(where)d Fi(P)2066 1994 y Fv(0)2126 2027 y Fp(is)f(the)i(MPP)g(of)g(the)f(TCG)h(of)f(the)609 2140 y(auxiliary)28 b(digraph,)h(then)h(go)h(to)g(Step)g(9.)382 2276 y Fg(Step)i(8.)46 b Fp(Put)30 b Fi(P)j Fp(:=)25 b Fi(P)1169 2244 y Fv(0)1223 2276 y Fp(and)30 b(go)h(to)g(Step)f(5.)382 2413 y Fg(Step)j(9.)46 b Fp(In)32 b(the)h(MPP)g Fi(P)1289 2380 y Fv(0)1313 2413 y Fp(,)h(\014nd)d(the)i(general)g(an)m(tipath)g Fi(U)2466 2380 y Fv(0)2489 2413 y Fp(.)48 b(If)33 b(this)e(an)m(tipath) i(con-)609 2526 y(tains)h(only)g(indep)s(enden)m(t)f(v)m(ertices,)j (then)f(put)f Fk(W)45 b Fp(=)32 b Fi(U)2602 2493 y Fv(0)2625 2526 y Fp(.)54 b(Otherwise,)34 b(go)i(to)609 2639 y(Step)30 b(11.)382 2775 y Fg(Step)j(10.)46 b Fp(Execute)37 b(the)f(cutting)f(op) s(eration)2009 2752 y Fk(~)1995 2775 y(G)p Fp(\()p Fk(W)13 b Fp(\))36 b(=)f Fk(\033)2430 2789 y Fj(W)2511 2775 y Fp(\()2560 2752 y Fk(~)2546 2775 y(G)p Fp(\()p Fk(X)2735 2742 y Fh(0)2775 2775 y Fp(\)\).)57 b(Remo)m(v)m(e)38 b(all)609 2888 y(\\considered")29 b(marks)f(of)h(arcs)g(of)1816 2865 y Fk(~)1803 2888 y(G)o Fp(\()p Fk(X)1991 2855 y Fh(0)2032 2888 y Fp(\),)g(put)2300 2865 y Fk(~)2286 2888 y(G)p Fp(\()p Fk(X)2475 2855 y Fh(0)2515 2888 y Fp(\))g(:=)2714 2865 y Fk(~)2700 2888 y(G)p Fp(\()p Fk(W)13 b Fp(\),)29 b(and)f(go)i(to)609 3001 y(Step)g(2.)382 3138 y Fg(Step)j(11.)46 b Fp(Construct)28 b(the)h(analyzing)e(digraph)2077 3115 y Fk(~)2064 3138 y(G)2136 3105 y Fv(\003)2174 3138 y Fp(\()p Fk(X)2291 3105 y Fh(0)2332 3138 y Fp(\))h(and)g(its)g (transitiv)m(e)f(closure)609 3251 y(graph)881 3228 y Fk(~)867 3251 y(G)939 3218 y Fv(\003)939 3273 y Fj(t)978 3251 y Fp(\()p Fk(X)1095 3218 y Fh(0)1135 3251 y Fp(\).)41 b(Find)29 b(an)h(MPP)h Fi(P)1886 3218 y Fv(\003)1956 3251 y Fp(of)2073 3228 y Fk(~)2060 3251 y(G)2132 3218 y Fv(\003)2132 3273 y Fj(t)2171 3251 y Fp(\()p Fk(X)2288 3218 y Fh(0)2328 3251 y Fp(\).)41 b(If)1522 3452 y Fk(C)7 b(ar)s(d)p Fp(\()p Fi(U)1834 3415 y Fv(\003)1873 3452 y Fp(\))26 b Fk(>)f(C)7 b(ar)s(d)p Fp(\()p Fk(X)2358 3415 y Fh(0)2397 3452 y Fp(\))p Fk(;)609 3654 y Fp(then)30 b(put)g Fk(W)38 b Fp(=)25 b Fi(U)1269 3621 y Fv(\003)1338 3654 y Fp(and)30 b(go)h(to)g(Step)f(10.)42 b(Otherwise,)29 b(go)j(to)f(Step)f(12.)382 3791 y Fg(Step)j(12.)46 b Fp(Find)36 b(an)i(o)s(dd)e(cycle)i Fk(C)1590 3806 y Fh(2)p Fj(k)r Fh(+1)1795 3791 y Fp(\()p Fk(k)j Fi(\025)c Fp(2\))h(of)f Fk(G)p Fp(,)i(restrictedly)d(stretc)m(hed)j(on)609 3904 y(paths)27 b(of)g(a)h(MPP)f(of)1375 3881 y Fk(~)1361 3904 y(G)1433 3918 y Fj(t)1462 3904 y Fp(\()p Fk(X)1579 3871 y Fh(0)1619 3904 y Fp(\),)i(and)d(in)m(tro)s(duce)g(the)h(co-v)m (ertex)j(to)e(the)f(curren)m(t)609 4017 y(digraph.)84 b(In)44 b(the)i(obtained)e(digraph)2073 3994 y Fk(~)2060 4017 y(G)o Fp(\()p Fk(X)2248 3984 y Fh(0)2289 4017 y Fp(\),)49 b(remo)m(v)m(e)e(all)d(\\considered")609 4130 y(marks)30 b(of)h(arcs)f(and)g(go)h(to)g(Step)f(2.)382 4266 y Fg(Step)j(13.)46 b Fp(If)33 b(the)g(digraph)1354 4243 y Fk(~)1340 4266 y(G)o Fp(\()p Fk(X)1528 4233 y Fh(0)1569 4266 y Fp(\))g(has)g(no)g(co-v)m(ertices,)j(stop)e(the)f (calculations,)g(as)609 4379 y(the)g(sough)m(t)g(partition)e Fq(F)h Fp(of)g Fk(G)g Fp(in)m(to)g(the)h(minim)m(um)c(n)m(um)m(b)s(er)i (of)i(cliques)e(has)609 4492 y(b)s(een)g(found.)42 b(Eac)m(h)32 b(elemen)m(t)g(of)g(MCP)f Fq(F)g Fp(is)f(formed)h(b)m(y)g(the)h(v)m (ertices)g(of)f(the)609 4605 y(corresp)s(onding)i(path)h(of)g(MPP)h Fi(P)42 b Fp(of)1989 4582 y Fk(~)1976 4605 y(G)2048 4619 y Fj(t)2077 4605 y Fp(\()p Fk(X)2194 4572 y Fh(0)2234 4605 y Fp(\).)53 b(If)34 b(the)h(digraph)2954 4582 y Fk(~)2940 4605 y(G)p Fp(\()p Fk(X)3129 4572 y Fh(0)3169 4605 y Fp(\))g(has)609 4718 y(at)c(least)g(one)g(co-v)m(ertex,)h(go)g (to)f(Step)f(14.)382 4854 y Fg(Step)j(14.)46 b Fp(Remo)m(v)m(e)35 b(all)e(co-v)m(ertices)i(in)d(the)h(digraph)2291 4831 y Fk(~)2278 4854 y(G)o Fp(\()p Fk(X)2466 4821 y Fh(0)2507 4854 y Fp(\).)49 b(The)33 b(paths)g(of)h(MPP)609 4967 y Fi(P)i Fp(of)822 4944 y Fk(~)809 4967 y(G)881 4981 y Fj(t)910 4967 y Fp(\()p Fk(X)1027 4934 y Fh(0)1067 4967 y Fp(\),)29 b(in)m(tersecting)f(other)g(v)m(ertices,)h(determine)f (the)g(required)e(MCP)609 5080 y(of)31 b(the)f(graph)g Fk(G)25 b Fp(=)g(\()p Fk(X)r(;)15 b Fp(\000\))26 b Fi(2)f Fk(L)1738 5094 y Fj(n)1785 5080 y Fp(.)382 5290 y Fq(Theorem)34 b(15)46 b Fg(The)31 b(algorithm)i(CLIP)-7 b(A)g(T-M)29 b(\014nds)j(a)f(p)-5 b(artition)33 b(of)e(the)h(gr)-5 b(aph)32 b Fk(G)25 b Fi(2)382 5403 y Fk(L)444 5417 y Fj(n)523 5403 y Fg(into)33 b(the)g(minimum)h(numb)-5 b(er)33 b(of)g(cliques)f(within)h(time)g Fk(O)s Fp(\()p Fk(n)2645 5370 y Fh(6)2684 5403 y Fp(\))p Fg(.)1831 5652 y Fp(35)p eop %%Page: 36 36 36 35 bop 523 548 a Fp(First)24 b(of)h(all,)g(notice)h(that)f(since)f (elemen)m(ts)i(of)f Fk(G)f Fi(2)h Fk(L)2364 562 y Fj(n)2436 548 y Fp(are)g(\014nite,)h(the)f(algorithm)382 661 y(CLIP)-8 b(A)g(T-M)35 b(is)f(\014nite,)i(and)e(constructs)h(an)g(MPP)g(of)g(the) g(TCG)g(of)g(some)g(digraph)395 751 y Fk(~)382 774 y(G)454 741 y Fv(0)509 774 y Fp(not)e(con)m(taining)f(mark)m(ed)h(v)m(ertices)g ({)g(see)h(Theorem)e(14.)48 b(Therefore,)34 b(according)382 887 y(to)46 b(Lemma)f(1,)50 b(the)45 b(existing)f(MPP)i(determines)e (the)h(partition)f(of)i(some)f(graph)382 1000 y Fk(G)454 967 y Fv(0)509 1000 y Fi(2)31 b Fk(L)663 1014 y Fj(n)p Fh(+)p Fj(p)835 1000 y Fp(in)m(to)j(the)h(minim)m(um)d(n)m(um)m(b)s(er) h(of)h(cliques.)52 b(Here,)36 b Fk(p)e Fp(is)g(the)g(n)m(um)m(b)s(er)f (of)382 1122 y(co-v)m(ertices)f(in)d(the)i(digraph)1437 1099 y Fk(~)1423 1122 y(G)1495 1089 y Fv(0)1518 1122 y Fp(\()p Fk(X)1635 1089 y Fh(0)1675 1122 y Fp(\).)523 1235 y(If)632 1212 y Fk(~)618 1235 y(G)690 1202 y Fv(0)747 1235 y Fp(do)s(es)j(not)h(con)m(tain)g(co-v)m(ertices,)j(that)d(is,)g Fk(p)d Fp(=)f(0,)37 b(then)d(the)g(found)g(MPP)382 1348 y(determines)26 b(some)i(MCP)g Fq(F)f Fp(of)h Fk(G)d Fi(2)f Fk(L)1741 1362 y Fj(n)1816 1348 y Fp(directly)-8 b(.)39 b(If)2275 1325 y Fk(~)2262 1348 y(G)2334 1315 y Fv(0)2357 1348 y Fp(\()p Fk(X)2474 1315 y Fh(0)2514 1348 y Fp(\))28 b(con)m(tains)g(co-v)m(ertices,)382 1461 y(then,)34 b(according)g(to)g(Theorem)f(8,)i(the)e(MCP)h(of)f Fk(G)d Fi(2)g Fk(L)2402 1475 y Fj(n)2482 1461 y Fp(ma)m(y)k(b)s(e)f (obtained)g(from)382 1574 y(the)d(MPP)h(of)893 1551 y Fk(~)879 1574 y(G)951 1541 y Fv(0)951 1596 y Fj(t)980 1574 y Fp(\()p Fk(X)1097 1541 y Fh(0)1138 1574 y Fp(\))f(b)m(y)h (deleting)e(existing)g(co-v)m(ertices.)523 1687 y(The)24 b(running)e(time)j(of)g(the)g(prop)s(osed)e(algorithm)h(is)g (determined,)h(ob)m(viously)-8 b(,)25 b(b)m(y)382 1800 y(the)30 b(time)h(of)f(the)h(construction)f(of)1644 1777 y Fk(~)1630 1800 y(G)1702 1767 y Fv(0)1725 1800 y Fp(\()p Fk(X)1842 1767 y Fh(0)1882 1800 y Fp(\),)h(whose)f(TCG)g(has)g(no)h (mark)m(ed)f(v)m(ertex.)523 1913 y(W)-8 b(e)28 b(ev)-5 b(aluate)28 b(the)f(op)s(erating)f(time)h(for)f(the)h(algorithm)f(CLIP) -8 b(A)g(T-M.)28 b(This)d(time)382 2026 y(is)k(determined)h(b)m(y)g (the)h(time)f(required)e(for)j(the)f(construction)h(of)f(a)h (VS-digraph)e(on)382 2139 y(Step)24 b(2.)40 b(Since)23 b(this)h(step)h(cannot)g(b)s(e)g(executed)g(more)g(than)g Fk(n)f Fp(times,)i(then)e(common)382 2251 y(running)k(time,)i(ob)m (viously)-8 b(,)30 b(is)f(equal)h(to)h Fk(n)20 b Fi(\002)g Fk(O)s Fp(\()p Fk(n)2143 2218 y Fh(5)2182 2251 y Fp(\))31 b(=)f Fk(O)s Fp(\()p Fk(n)2511 2218 y Fh(6)2550 2251 y Fp(\).)453 b(Q.E.D.)382 2538 y Fl(12)135 b(Ac)l(kno)l(wledgemen)l(ts) 382 2741 y Fp(I)33 b(w)m(ould)e(lik)m(e)h(to)i(thank)e(the)i(writer)d (Brian)h(Wizard)h(for)f(his)g(supp)s(ort)f(of)i(this)f(w)m(ork)382 2854 y(and)f(Marsha)h(Calhoun)e(for)i(editing)e(the)i(English)d(v)m (ersion)j(of)g(this)e(pap)s(er.)44 b(I)32 b(w)m(ould)382 2967 y(lik)m(e)24 b(also)i(to)g(thank)f(Stas)g(Busygin)f(for)h(useful)f (con)m(v)m(ersations)i(and)f(a)g(help)f(in)g(editing)382 3080 y(the)30 b(English)e(v)m(ersion.)382 3366 y Fl(References)382 3569 y Fp([1])46 b(P)-8 b(.)40 b(Crescenzi)f(and)f(V.)i(Kann.)66 b(A)40 b(comp)s(endium)c(of)k(NP)f(optimization)f(prob-)523 3682 y(lems.)52 b(T)-8 b(ec)m(hnical)34 b(rep)s(ort,)h(Ro)m(y)m(al)g (Institute)e(of)i(T)-8 b(ec)m(hnology)g(,)37 b(Sto)s(c)m(holm,)e(1998.) 523 3795 y(This)48 b(is)g(the)h(catalog)i(of)e Fq(NP)g Fp(optimization)f(problems.)g(Also)g(a)m(v)-5 b(ailable)49 b(at)523 3908 y(ftp://ftp.nada.kth.se/Theory/Viggo-Kann/comp)s (endium.ps.)382 4095 y([2])d(L.)37 b(R.)g(F)-8 b(ord)37 b(and)f(D.)i(R.)f(F)-8 b(ulk)m(erson.)59 b Fg(Flows)40 b(in)f(Networks)p Fp(.)60 b(Princeton)37 b(Univ.)523 4208 y(Press,)30 b(Princeton,)g(N.)h(J.,)g(1962.)382 4396 y([3])46 b(M.)63 b(R.)f(Garey)h(and)e(D.)i(S.)f(Johnson.)134 b Fg(Computers)64 b(and)e(Intr)-5 b(actability)p Fp(.)523 4509 y(W.H.F)d(reeman)33 b(and)c(Compan)m(y)-8 b(,)31 b(San)f(F)-8 b(rancisco,)31 b(1979.)382 4696 y([4])46 b(J.)23 b(E.)g(Hop)s(croft)g(and)f(R.)g(M.)i(Karp.)j(A)c Fk(n)1918 4663 y Fh(5)p Fj(=)p Fh(2)2050 4696 y Fp(algorithm)f(for)g (maxim)m(um)g(matc)m(hing)523 4809 y(in)29 b(bipartite)g(graphs.)40 b Fg(J.)32 b(SIAM)g(Comp.)p Fp(,)g(2:225{231,)i(1973.)382 4997 y([5])46 b(C.)33 b(H.)h(P)m(apadimitriou)c(and)j(K.)g(Steiglitz.) 48 b Fg(Combinatorial)37 b(optimization:)50 b(A)n(l-)523 5110 y(gorithms)44 b(and)f(Complexity)p Fp(.)72 b(Pren)m(tice-hall)40 b(Inc.,)j(Engiew)m(o)s(o)s(d)c(Cli\013s,)j(N.)e(J.,)523 5223 y(1982.)1831 5652 y(36)p eop %%Page: 37 37 37 36 bop 382 548 a Fp([6])46 b(A.)h(D.)h(Plotnik)m(o)m(v.)89 b(P)m(olynomial-time)45 b(partition)g(of)i(a)g(graph)f(in)m(to)h (cliques.)523 661 y Fg(SouthWest)34 b(Journal)g(Pur)-5 b(e)32 b(and)i(Applie)-5 b(d)33 b(Mathematics)p Fp(,)f(1:16{21,)i (1996.)382 848 y([7])46 b(F.)c(M.)f(Reingold,)h(J.)f(Niv)m(ergelt,)j (and)c(N.)h(Deo.)73 b Fg(Combinatorial)45 b(A)n(lgorithms)523 961 y(\(The)-5 b(ory)32 b(and)g(Pr)-5 b(actic)g(e\))p Fp(.)37 b(Pren)m(tice-hall)27 b(Inc.,)i(Engiew)m(o)s(o)s(d)e(Cli\013s,) g(N.)h(J.,)h(1977.)382 1149 y([8])46 b(M.)c(N.)f(S.)g(Sw)m(am)m(y)g (and)g(K.)g(Th)m(ulasiraman.)70 b Fg(Gr)-5 b(aphs,)47 b(Networks)c(and)g(A)n(lgo-)523 1262 y(rithms)p Fp(.)f(John)30 b(Wiley)f(&)h(Sons,)g(N.)h(Y.,)g(Chic)m(hester,)f(1981.)382 1450 y([9])46 b(D.)36 b(B.)f(W)-8 b(est.)56 b Fg(Intr)-5 b(o)g(duction)39 b(to)f(Gr)-5 b(aph)39 b(The)-5 b(ory)p Fp(.)55 b(Pren)m(tice)35 b(Hall,)h(Inc.,)g(Upp)s(er)523 1562 y(Saddle)29 b(Riv)m(er,)h(NJ,)h(1996.)1831 5652 y(37)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF