Characterization and Construction of Permutation Graphs

Transcription

1 Ope Joural of Dscrete Mathematcs, 03, 3, Publshed Ole Jauary 03 ( Characterzato ad Costructo of Permutato Graphs Seero V Geraco, Teofa A Rapaut, Phoebe Chloe F Ramos Departmet of Mathematcs, De La Salle Uersty, Mala, Phlppes Departmet of Mathematcs ad Computer Scece, Uersty of the Phlppes, Baguo Cty, Phlppes Emal: seerogeraco@dlsueduph Receed September 5, 0; resed October 5, 0; accepted October 5, 0 ABSTRACT If s a permutato of,,,, the graph G has ertces,,, where xy s a edge of G f ad oly f x, y or y, x s a erso of Ay graph somorphc to G s called a permutato graph I 967 Galla characterzed permutato graphs terms of forbdde duced subgraphs I 97 Puel, Lempel, ad Ee showed that a graph s a permutato graph f ad oly f both the graph ad ts complemet hae traste oretatos I 00 Lmouzy characterzed permutato graphs terms of forbdde Sedel mors I ths paper, we characterze permutato graphs terms of a cohese order of ts ertces We show that oly the caterpllars are permutato graphs amog the trees A smple method of costructg permutato graphs s also preseted here Keywords: Permutato; Ierso; Permutato Graph; Cohese Order; Oreted Graph; Touramet Score Sequece; Caterpllar; Graph Composto Itroducto A becto of,,, to tself s called a permutato of order We shall wrte a, a,, a to mea that a for,,, We shall deote by the set of all permutatos of,,, A erso of s a ordered par a, a where but a a Equaletly, x, y s a erso f ad oly f x y ad x y Defto Let The graph of ersos of, deoted by G, s the graph wth ertces,,, where xy s a edge of G f ad oly f x, y or y, x s a erso of The term graph of ersos was used by Ramos [] For our purpose ths paper, ay graph somorphc to G for some permutato wll be called a permutato graph There s a mplemetato PermutatoGraph[p] the Combatorca package of Mathematca [] that creates the permutato graph G p I 967 Galla [3] characterzed permutato graphs terms of forbdde duced subgraphs I 97 Puel, Lempel, ad Ee [] showed that a graph G s a permutato graph f ad oly f both G ad ts complemet G hae traste oretatos Recetly 00 Lmouzy [5] gae a characterzato of permutato graphs terms of forbdde Sedel mors A characterzato of permutato graphs terms of cohese ertex-set order s establshed ths paper We proe that the oly permutato graphs amog the trees are the caterpllars I addto, we descrbe a smple method of costructg permutato graphs Cohese Vertex-Set Order The ertex-set of a graph G wll be deoted by V G whle the edge-set wll be deoted by EG A edge wth ed-ertces a ad b wll be deoted by ab or ba For graph theoretc terms used here wthout defto, the book by Harary [6] may be referred to Cosder the permutato 3,,,6,5, The ersos of are 3,, 3,,,, 6,5, 6,, ad 5, It s coeet to draw the graph G wth the ertces a le followg ther arragemet A drawg of G s show Fgure There are some mportat propertes of the drawg that we eed to take ote of (a) If ab ad bc are two edges of the graph where b les betwee a ad c the drawg, the ac s also a edge Fgure Permutato graph G,,,,,, σ σ 365 Copyrght 03 ScRes

2 3 S V GERVACIO ET AL (b) If u s a edge ad x s a ertex that les betwee u ad the drawg, the ether u s a edge or x s a edge We defe more precsely the propertes that we obsered Defto Let G be a graph of order A arragemet,,, of the ertces of G s called a cohese ertex-set order of G (or smply cohese order G ) f the followg codtos are satsfed: (a) If k ad k, k EG, the EG (b) If k ad EG, the k EG or k VG The c omplemet of a graph G, deoted by G has the same ertex-set as G ad two dstct ertces a ad b form the edge ab G f ad oly f ab s ot a edge G Lemma Let G be a g raph The s a cohese order of G f ad oly f s a cohese order of G Proof Let,,, be a cohese order of G We clam that the same s a cohese order of G To proe a for G, let k ad k be ertces of G such that k The k ad k are ot edges G By property b of a cohese order, the edge s ot G Hece, s a edge of G To proe b for G, l et be a edge of G wth Let k be a teger su ch that k Sce s G, the t s ot G By property a of a cohese order (for G ) the edges k ad c k aot be both G He ce at least oe of them s G The coerse follows sce G G The ext result follows eas ly from the defto of permutato graph ad cohese order We shall omt the proof of ths theorem Theorem Let The,,, s a cohese order of the permutato gra ph G Note that,,, s a cohese orde r of a graph G f ad oly f,,, s a cohese order of G To assg a drecto to a edge ab of a graph G meas to chage ab to ether the ordered par ab, or the ordered par b, a Defto A oretato of a graph G s the dgraph obtaed by assgg a drecto to each edge of G The drected edges, whch are ordered pars, are called arcs A dgraph D s sad to be traste f x, z s a arc of D wheeer x, y ad y, z are arcs D I a dgraph D, the out-degree of a ertex x, deoted by deg D x or smply deg x s the umbe r of ertce s y D such that x, y s a arc D The -degr ee of x, deoted by deg D x or deg x s the umber of ertces z D such that zx, s a arc D A oreted complete graph s called a tou ramet [7] The score of a ertex x a touramet, deoted by s x s defed by s x deg x The score sequece of a touramet s the sequece of scores arraged o-decreasg order The followg theorem [8] s ot dffcult, ad s stated wthout proof Theorem Let T be a touramet of order The followg statemets are equalet: ) T s trast e ) For all ertces x ad y T, f x, y s a arc of T, the s x s y 3) For all ertces x ad y T, f s x s y, the x, y s a arc of T ) Th e score sequece of T s 0,,,, Our ma result, whch characterzes permutato graphs, s the followg theorem Theorem 3 A graph G s a permut ato graph f ad oly f t has a cohese order Proof If G s a permutato graph, the G s somorphc to G for som e permutato By Theorem, s a a cohese order of G Let be a somorphs m of G to G The s a cohese order of G Coersely, let G be a graph wth a cohese order,,, Oret G t o obta a dgraph D as follows: For each edge G, assg the drecto, f ; otherwse assg the drecto, of a cohese order, t follows that By property a D s traste Exted D to a touramet by oretg the complemet G of G as follows: If but s ot D, assg the drecto, to the edge G By Lemma s a cohese order of G So lkews e, the oretato of G obtaed ths maer s also traste Let us deote ths dgrap h by D The uo of D ad D s a oretato of G G Sce G G s complete, the T D D s a touramet We clam that T s a traste touramet Let x, y ad y, z be arc s of T If both arcs be log to D or to D, the x, z s T because both D ad D are traste So let us assume that oe of the arcs be log to D ad the other arc belog to D Wthout loss of geeralty, assume that x, y s a arc D, a d y, z s a arc D If x, z s D, we are doe If x, z s ot D, the zx, s D Sce D s traste ad zx,, y, z are D, the y, x s D Ths s a cotradcto because x, y s D Copyrght 03 ScRes

3 S V GERVACIO ET AL 35 B y Theorem, the score sequece of T s 0,,, Let be the permutato defed by s, where s s the score of T We clam that the mappg : s s a somorphsm of G to G The mappg s becte sce the scor es of the ertces are dstct It remas to show that pre- seres adacecy Let be a edge of G, where < I D we hae the arc, Sce the touramet T s traste, the by Theorem, s s Hece,, s a erso of Therefore, ad are adacet G Coersely, let ab be a edge G The ether ab, or ba, s a erso Wthout loss of geeralty, assume that ab, s a erso Let a s ad b s, where Sce ab, s a erso, we hae a b, or s s Therefore, the arc, s T Sce, the arc, must be D Cosequetly, the edge s G Here s a llustrato of the costructe proof of Theorem 3 Cosder the graph G show Fgure wth a cohese order x, x, x, x 3, x 5 To be able to follow the dscusso the proof of theorem wthout dffculty, let,,,, x, x, x, x, x Usg the bottom drawg Fgure, we costruct a dgraph by drectg all edges from left to rght For two ertces ot adacet G, we assg the arc that goes from rght to left The the result s a traste touramet It s ot dffcult to get the score of ay ertex ths touramet We smp ly cout the eastboud arcs ad the westboud arcs wth a fxed tal Cosder for example, x The umber of eastboud arcs wth tal at x s 3 The umber of westboud arcs s smply the umber of ertces to ts left that are ot adacet to to x The table below summarzes the scores of the ertces Vertex = x = x 3 = x = x 3 5 = x 5 Score, s( ) 0 3 Take the pe o = s Th e 3, 5,,, The permutato graph G s show Fgure 3 rmutat defed by 3 Costructo ad Examples of Permutato Graphs Some fudametal facts about permutato graphs are ge the ext theorem Theorem 3 Let G be a graph The followg are equalet: (a) G s a permutato graph (b) G s a permutato graph G x x x 3 x x 5 x Fgure A graph G wth cohese order,,,, x, x, x, x, x x x x 3 x G, 35,,,, Fgure 3 The permutato graph (c) Eery duced subgraph of G s a permutato graph (d) Eery coected compoet of G s a permutato graph Proof From Lemma, G has a cohese order f ad oly f G has a cohese order Therefore, (a) ad (b) are equalet If,,, s a cohese order of G, the the su bgraph of G duced by a set of ertces,,, k, where k has cohese order,,, k ad therefore s a permutato graph Hece, (a) ad ( c) are equalet Statemet (c) mples statemet (d) because a coected compoet of G s a duced subgraph of G It remas to show that (d) mples ay of (a), (b), (c) Let G hae coected compoe ts G, G,, Gk ad let be the order of G Let,,, be a cohese order of G The,,, k,,,,,,,,,,,, k k k k s a cohese order o f G Therefore G s a permutato graph We ca ow detfy permutato graphs through the exstece of a cohese order Moreoer, we ca ee determe a permutato that geerates a permutato graph somorphc to the graph hag a cohese order Paths P ad stars K, are permutato graphs because they hae cohese orders as llustrated Fgure I the drawg of the path P, we hae,,, 3, 6, etc 3 5 Codto (a) s acuously satsfed because there s o σ Copyrght 03 ScRes

4 36 S V GERVACIO ET AL par of arcs, ad k such that k Note fo r example that,3 s a a rc ad the ertces ad are betwee ad 3 the drawg W e hae adacet to ad adacet to 3 Ths llustrates codto (b) I the drawg of the star K, we see that for eery arc 0, k where k all ertces wth 0 k are betwee 0 ad k Moreoer, the ertex s adacet to 0 Therefore codto (b) s satsfed Codto (a) s satsfed acuously Paths ad stars are trees but ot all trees are permutato graphs Cosder the tree K,3 formed by subddg each edge of the star K,3 to two edges, as show Fgure 5 It s ot dffcult to argue drectly that K,3 has o cohese order Therefore ths s ot a permutato graph Ths result s also establshed by Lmouzy [5] where he used the symbol T for K,3 Harary ad Schwek [9] defed a caterpllar to be a tree wth the property that the remoal of all pedat ertces results to a path Fgure 6 shows a caterpllar wth 5 pedat ertces The remoal of these 5 pedat ertces yelds the path P 8 The ext lemma s easy ad ts proof s omtted Lemma 3 A tree s a caterpllar f ad oly f t does ot cota K,3 as a subgraph Theorem 3 A tree s a permutato graph f ad oly f t s a caterpllar Proof A tree that cotas K,3 s ot a permutato Path, P Star, K, Fgure Cohese order of paths ad stars Star, K,3 Fgure 5 The tree of K 3, K 3, obtaed by subddg the edges Fgure 6 A caterpllar wth 5 pedat ertces K *,3 graph because K,3 s ot a permutato graph Therefore, all we eed to show s that a caterpllar s a permutato graph Let C be a caterpllar ad let P be the path obtaed from C by remog the pedat ertces If, the C s ether the tral graph or the star K, p for some p Sce the tral graph ad the stars are permutato graphs, we assume that Let us form the cohese order of P as show Fgure Le t S be a set of pedat ertces of C all adacet to the same ertex of P If s odd, we sert the ertces S mmedately to the left of the ertex of the path (see Fgure ) If s ee we sert the ertces S betwee ad The result s a cohese order of C Therefore C s a permutato graph Defto 3 Let G be a graph wth ertces x, x,, x ad let H, H,, H be a collecto of arbtrary graphs The composto by G of H, H,, H, deoted by GH, H,, H s the graph formed by takg the dsot uo of the graphs H ad the addg all edges of the form ab where a s H, b s H wheeer x x s a edge of G If each H s equal to a fxed grap h H, we use the symbol GH to deote the composto The sum of two graphs L ad M, deoted by L M s formed by takg the dsot uo of L ad M ad the addg all edges of the for m ab where a V L ad b VM Thus, the composto GH, H,, H s formed by takg the dsot uo of the graphs H ad the formg the sum H H f the assocated ertces x ad x of G are adacet Theorem 33 Let G be a graph of order ad let H, H,, H be arbtrary graphs The G H, H,, H s a permutato graph f ad oly f G, H, H,, H are permutato graphs Proof Frst, assume that GH, H,, H s a permutato graph Each graph H s a duced subgraph of GH, H,, H Therefore, each H s a permutato graph If we take a ertex x from each H, the the subgraph duced by these ertces s so- morphc to G Therefore G s a permutato graph Coersely, assume that G, H, H,, H are all permutato graphs The there s a cohese order,,, of G Let be the order of H The the ertces of H has a cohese order x, x,, x It s easy to check that,,, s a cohe- se order of G H, H,, H Copyrght 03 ScRes

5 S V GERVACIO ET AL 37 Theorem 33 actually ges us a easy way of costructg permutato graphs by composto To llustrate ths, let G be the star K,3 wth cetral ertex x ad pedat ertces x, x3, x, the G P C, P, P, 3 3 s show Fgure 7 All graphs of order at most are permutato graphs [] Therefore, GP, C3, P, P 3 s a permutato graph Eery graph G of order may be wrtte as G G P, P,, P ad G K G If these are the oly ways G ca be wrtte as a composto, the we say that G s prme It s easy to see that amog the complete graphs, oly K ad K are prme permutato graphs Amog trees wth dameter ot exceedg 3, t s easy to check that oly the paths P, P, ad P are prme permutato graphs These are all caterpllars that do ot hae two pedat ertces adacet to a commo ertex Note that P 3 whch s excluded from the lst s a caterpllar wth two pedat ertces hag a commo eghbor Theorem 3 A tree s a prme permutato graph f ad oly f t s a caterpllar where o two pedat ertces hae a commo eghbor Proof I ew of our obserato about trees wth dameter ot exceedg 3, we assume throughout that T has dameter at least Let T be a tree of order Assume that T s a prme permutato graph By Theorem 3 T s a caterpllar Suppose that x ad x are pedat ertces wth a commo eghbor y Let G be the tree obtaed from T by detfyg x ad x Let y, y,, y be the ertces of G Wthout loss of geeralty, assume that y s the ertex resultg from the detfcato of x ad x Let H be the grap h wth two ertces but wthout a edge, ad let H be the tral graph for, 3,, The T G H, H,, H Ths cotradcts the fact that T s prme Fgure 7 The composto by K 3, of P, C, P, P 3 3 Coersely, assume that T s a caterpllar wth o two pedat ertces hag a commo eghbor Suppose that T s a ot a prme permutato graph The for some o-tral graph G wth ertces y, y,, yk, T GH H H,,, k Wthout loss of geeralty, we may assume that H cotas at least two ertces Now, G must be co- ected for otherwse, T s dscoect ed Let y be adacet to y wthout loss of geeralty The H H s a subgraph of T If H has at least two ertces, the there wll be a cycle H H Therefore, H has oly oe ertex I G, y caot be adacet aymore to ay other ertex for otherwse, we would also create a cycle of legth Now cosder H H There caot be adacet ertces H for otherwse we wll create a cycle of legth 3 But the all ertces H are pedat ertces of T ad they hae a commo eghbor, the ert ex H Ths s a cotradcto Theorem 35 Let G b e a decomposable permutato graph The there exsts a o-tral prme permutato graph U ad permutato grap hs H, H,, Hk whch are subgraphs of G such that G UH, H,, Hk Proof Let G U H, H,, Hk be a decomposto of G, where U s o-tral If we take oe ertex x from each H, the the subgraph duced by these ertces s somorphc to U Hece, U must be a permutato graph Each H s a duced subgraph of UH, H,, H k The refore, each H s a perm utato graph Assume that U has smallest order am og all such decompostos of G We clam that U s a prme permutato graph S uppose that U s ot prme Let U V L, L,, Lp be a decomposto of U, where V s o-tral Sce V s a decomposto of U ad ertces of U are the duced subgraphs H the each L s a assocated wth subset of H, H,, Lk We may assume that L s a duced subgraph of G Hece G V L, L,, Lp But ths cotradcts the choce of U Therefore, U must be a prme permutato graph Cocludg Remarks Theore m 35 s a far structural descrpto of a per- Copyrght 03 ScRes

6 38 S V GERVACIO ET AL mutato graph Each H the decomposto G U H, H,, Hk s a permutato graph ad so s tself prme permutato graph or a composto of permutato graphs by a prme permutato graph So we see that a permutato graph s expressble terms of prme permutato graphs by compostos We hae determed already the prme permutato trees, ge Theorem 3 Oe terestg problem to cosder s the characterzato of prme permutato graphs REFERENCES [] P C F Ramos, O Graphs of Iersos of Permutatos, Master s Thess, Uersty of the Phlppes, Baguo Cty, 0 [] S Skea, Implemetg Dscrete Mathematcs: Combatorcs ad Graph Theory wth Mathematca, Addso- Wesley, Readg, 990 [3] T G alla, Trast Oreterbare Graphe, Acta Mathe- matca Academae Scetarum Hugarca, Vol 8, No -, 967, pp 5-66 do:0007/bf00096 [] A Puel, A Lempel ad S Ee, Traste Oretato of Graphs ad Idetfcato of Permutato Graphs, Caada Joural of Mathematcs, Vol 3, No, 97, pp do:053/cjm [5] V Lmouzy, Sedel Mor, Permutato Graphs ad Combatoral Propertes, I: Lecture Notes Computer Scece Volume 6506, Sprger, Berl, 00, pp 9-05 [6] F Harary, Graph Theory, Addso-Wesley Publshg Compay, Bosto, 969 [7] J W Moo, Topcs o Touramets, Holt, Rehart ad Wsto, New York, 968 [8] S V Geraco, Touramet Score Sequeces, Aals of the New York Academy of Sceces, Vol 576, Graph Theory ad Its Applcatos, East ad West: Proceedgs of st Cha-USA Iteratoal Graph Theory Coferece, Ja, Cha, Jue 986, pp 00-0 [9] F Harary ad A J Schwek, The Number of Caterpllars, Dscrete Mathematcs, Vol 6, No, 973, pp do:006/00-365x(73) Copyrght 03 ScRes