Planar Graph Isomorphism is in Log-Space

Transcription

1 Plnr Grph Isomorphism is in Log-Spce Smir Dtt 1 Nutn Limye 2 Prjkt Nimbhorkr 2 Thoms Thieruf 3 Fbin Wgner 4 1 Chenni Mthemticl Institute sdtt@cmi.c.in 2 The Institute of Mthemticl Sciences {nutn,prjkt}@imsc.res.in 3 Fk. Elektronik und Informtik, HTW Alen 4 Institut für Theoretische Informtik, Universität Ulm, Ulm {thoms.thieruf,fbin.wgner}@uni-ulm.de June 15, 2009 Abstrct Grph Isomorphism is the prime exmple of computtionl problem with wide difference between the best known lower nd upper bounds on its complexity. There is significnt gp between extnt lower nd upper bounds for plnr grphs s well. We bridge the gp for this nturl nd importnt specil cse by presenting n upper bound tht mtches the known log-spce hrdness [JKMT03]. In fct, we show the formlly stronger result tht plnr grph cnoniztion is in log-spce. This improves the previously known upper bound of AC 1 [MR91]. Our lgorithm first constructs the biconnected component tree of connected plnr grph nd then refines ech biconnected component into triconnected component tree. The next step is to log-spce reduce the biconnected plnr grph isomorphism nd cnoniztion problems to those for 3-connected plnr grphs, which re known to be in log-spce by [DLN08]. This is chieved by using the bove decomposition, nd by mking significnt modifictions to Lindell s lgorithm for tree cnoniztion, long with chnges in the spce complexity nlysis. The reduction from the connected cse to the biconnected cse requires further new ides, including non-trivil cse nlysis nd group theoretic lemm to bound the number of utomorphisms of colored 3-connected plnr grph. This lemm is crucil for the reduction to work in log-spce. A preliminry version of the pper ppered t rxiv: v1 nd rxiv: v2. Supported by DFG grnts Scho 302/7-2 nd TO 200/2-2. Supported by DFG grnts Scho 302/7-2 nd TO 200/2-2. 1

2 1 Introduction The grph isomorphism problem GI consists of deciding whether there is bijection between the nodes of two grphs, which preserves the djcency reltions. The wide gp between the known lower nd upper bounds hs kept live the reserch interest in GI. The problem is clerly in NP, nd, by group theoretic proof, lso in SPP [AK06]. This is the current frontier of our knowledge s fr s upper bounds go. The inbility to give efficient lgorithms for the problem would led one to believe tht the problem is provbly hrd. NP-hrdness is precluded by result tht sttes if GI is NP-hrd then the polynomil time hierrchy collpses to the second level [BHZ87, Sch88]. Wht is more surprising is tht not even P-hrdness is known for the problem. The best we know is tht GI is hrd for DET [Tor04], the clss of problems NC 1 -reducible to the determinnt, defined by Cook [Coo85]. While this enormous gp hs motivted study of isomorphism in generl grphs, it hs lso induced reserch in isomorphism restricted to specil cses of grphs where this gp cn be reduced. Tournments re n exmple of directed grphs where the DET lower bound is preserved [Wg07], while there is qusi-polynomil time upper bound [BL83]. Trees re n exmple of grphs where the lower nd upper bounds mtch nd re L [Lin92]. Note tht for trees, the problem s complexity crucilly depends on the input encoding: if the trees re presented s strings then the lower nd upper bound re NC 1 [MJT98, Bus97]). Lindell s log-spce result hs been extended to prtil 2-trees, lso known s generlized series-prllel grphs [ADK08]. Trees nd prtil 2-trees re specil cses of plnr grphs. In this pper we consider plnr grph isomorphism nd settle its complexity by significntly improving the known upper bound of AC 1. The result is prticulrly stisfying becuse Plnr Grph Isomorphism turns out to be complete for well-known nd nturl complexity clss, nmely log-spce: L. Plnr Grph Isomorphism hs been studied in its own right since the erly dys of computer science. Weinberg [Wei66] presented n O(n 2 ) lgorithm for testing isomorphism of 3-connected plnr grphs. Hopcroft nd Trjn [HT74] extended this to generl plnr grphs, improving the time complexity to O(n log n). Hopcroft nd Wong [HW74] further improved it to O(n). Recently Kukluk, Holder, nd Cook [KHC04] gve n O(n 2 ) lgorithm for plnr grph isomorphism, which is suitble for prcticl pplictions. The prllel complexity of Plnr Grph Isomorphism ws first considered by Miller nd Reif [MR91] nd Rmchndrn nd Reif [RR94]. They showed tht the upper bound is AC 1, see lso [Ver07]. Recent work hs delt with further specil cse, nmely 3-connected plnr grphs. Thieruf nd Wgner [TW08] presented new upper bound of UL coul, mking use of the mchinery developed for the rechbility problem [RA97] nd specificlly for plnr rechbility [ADR05, BTV07]. They lso show tht the problem is L-hrd. Further progress, in the form of log-spce lgorithm is mde by Dtt, Limye nd Nimbhorkr [DLN08] where the 3-connected plnr cse is settled, by building on ides from [TW08] nd using Reingold s construction of universl explortion sequences [Rei05]. We summrise the known results for plnr grphs nd their restrictions s follows: Grph clss Lower bound Upper bound Trees L [MJT98] L [Lin92] Prtil 2-trees L L [ADK08] 3-connected plnr grphs L [TW08] L [DLN08] Plnr grphs L AC 1 [RR94] 2

3 The current work is nturl culmintion of this series where we settle the complexity question for plnr grph isomorphism by presenting the first log-spce lgorithm for the problem. In fct, we give log-spce lgorithm for the grph cnoniztion problem, to which grph isomorphism reduces. The cnoniztion involves ssigning to ech grph n isomorphism invrint, polynomil length string. We consider plnr undirected grphs without prllel edges nd loops, lso clled simple grphs. For plnr grphs tht re not simple there re log-spce mny-one reductions to simple plnr grphs (cf. [KST93]). Our lgorithm consists of the following steps. 1. Decompose the plnr grph into its biconnected components nd construct biconnected component tree in log-spce ([ADK08], cf. [TW09] nd Section 3). 2. Decompose biconnected plnr components into their triconnected components to obtin triconnected component tree in log-spce. This is essentilly prllel implementtion of the sequentil lgorithm of [HT73] (Section 3). 3. Invoke the lgorithm of Dtt, Limye nd Nimbhorkr [DLN08] to cnonize the triconnected components of the grph. 4. Cnonize biconnected plnr grphs using their triconnected component trees. Lindell s lgorithm [Lin92] for tree cnoniztion nd its complexity nlysis hd to be modified in non-trivil wy for this step to work in log-spce (Section 4), i.e. from [Lin92] to their triconnected component trees. 5. Cnonize plnr grphs using their biconnected component trees, by substituting the biconnected components with their triconnected component trees (Section 5). Notice, tht in Step 4, pirwise isomorphism of two triconnected component trees lbelled with the cnons of their components does not imply isomorphism of the corresponding grphs. Figure 1 illustrtes this fct. So, nïve combintion of [Lin92] nd [DLN08] does not work. We need to introduce the concept of orienttions of seprting pirs (see Section 4.2 for detils) to ensure the extendibility of isomorphism of individul 3-connected components to the entire biconnected plnr grph. G H S {,b} G 0 b b G 1 G 2 b b b b T {,b } b H 0 b H 1 b H 2 b Figure 1: The grphs G nd H hve the sme triconnected component trees but re not isomorphic. 3

4 Triconnected components hve t most two embeddings on the sphere (cf. [Whi33]). This property helps to perform Step 4 in L. Biconnected components do not hve this property. Hence, the nïve pproch for Step 5 would need to keep trck of exponentilly mny cses. We solve this problem by resorting to n intricte cse nlysis, nd group theoretic lemm (Lemm 5.3) to bound the number of utomorphisms of coloured 3-connected plnr grph. We emphsize tht Step 5 does not use Step 4 s blck-box but s co-routine. In fct, we mintin two (logrithmiclly bounded) work-tpes for the two steps. 2 Definitions nd Nottions We recll some bsic grph theoretic notions. A grph G = (V,E) is connected if there is pth between ny two vertices in G. We consider (,b) s n ordered pir nd {,b} without ny order of nd b. For U V let G(U) be the induced subgrph of G on U. A vertex v V is n rticultion point if G(V \{v}) is not connected. A pir of vertices u,v V is seprting pir if G(V \ {u,v}) is not connected. A biconnected grph contins no rticultion points. A 3-connected grph contins no seprting pirs. A triconnected grph is either 3-connected grph or cycle or 3-bond. A k-bond is grph consisting of two vertices joined by k edges. A pir of vertices {,b} is sid to be 3-connected if there re three or more vertex-disjoint pths between them. For node v let d(v) be the mximl distnce tht v hs to ny of the other nodes of G. Let C be the set of nodes v of G tht hve miniml vlue d(v). The set C is clled the center of G. In other words, vertices in the center minimize the mximl distnce from other vertices in the grph. Note tht if G is tree such tht every pth from leve to leve hs even length, then the center consists of only one node, nmely the midpoint of longest pth in the tree. Let E v be the set of edges incident to v. A permuttion ρ v on E v tht hs only one cycle is clled rottion. A rottion scheme for grph G is set ρ of rottions, ρ = {ρ v v V nd ρ v is rottion on E v }. Let ρ 1 be the set of inverse rottions, ρ 1 = {ρ 1 v v V }. A rottion scheme ρ describes n embedding of grph G in the plne. If the embedding is plnr, we cll ρ plnr rottion scheme. Note tht in this cse ρ 1 is plnr rottion scheme s well. Allender nd Mhjn [AM00] showed tht plnr rottion scheme for plnr grph cn be computed in log-spce. Two grphs G 1 = (V 1,E 1 ) nd G 2 = (V 2,E 2 ) re sid to be isomorphic (G 1 = G2 ) if there is bijection φ : V 1 V 2 such tht {u,v} E 1 if nd only if {φ(u),φ(v)} E 2. Grph isomorphism (GI) is the problem of deciding whether two given grphs re isomorphic. A plnr grph G, long with its plnr embedding (given by ρ) is clled plne grph Ĝ = (G, ρ). A plne grph divides the plne into regions. Ech such region is clled fce. Let Plnr-GI be the specil cse of GI when the given grphs re plnr. The biconnected (respectively, 3-connected) plnr GI is specil cse of Plnr-GI when the grphs re biconnected (3-connected) plnr grphs. Let G be clss of grphs. Let f : G {0,1} be function such tht for ll G,H G we hve G = H f(g) = f(h). Then f computes complete invrint for G. If f computes for G grph f(g) such tht G = f(g) then we cll f(g) the cnon for G. By L we denote the lnguges computble by log-spce bounded Turing mchine. 4

5 3 Isomorphism Order of Connected Plnr Grphs In this section, we give proof for the following theorem. Theorem 3.1 The decomposition of biconnected plnr grphs into triconnected components is in log-spce. Hopcroft nd Trjn [HT73] presented sequentil lgorithm for the decomposition of biconnected plnr grph into its triconnected components. Their lgorithm recursively removes seprting pirs from the grph nd puts copy of the seprting pir in ech of the components so formed. The nodes in the seprting pir re connected by virtul edge. If simple cycles re split t ny intermedite steps then they re combined lter. This gives decomposition which is unique [Mc37]. We describe log-spce lgorithm for such decomposition of biconnected plnr grph. We strt with definitions nd then prove some properties of seprting pirs. Definition 3.2 In plne grph Ĝ, seprting pir {,b} is sid to spn fce f if both its endpoints,b lie on the boundry of f. Let v 0,v 1,...,v k be fce boundry. Two seprting pirs {v i,v j }, {v i,v j } re clled intersecting if i < i < j < j, nd non-intersecting otherwise. Lemm 3.3 Every seprting pir spns some fce. To see this, note tht in plne grph Ĝ, split component of seprting pir is embedded in some fce. This cn be considered s the spnned fce. A seprting pir {,b} tht spns fce f is clled 3-connected if there re t lest three vertex-disjoint pths between,b i.e. there is pth between,b in Ĝ which is vertex-disjoint from the boundry of f. The following lemm enbles us to remove ll the 3-connected seprting pirs simultneously. Lemm 3.4 In plne grph non-intersecting. Ĝ, 3-connected seprting pirs which spn the sme fce re Proof. Suppose {,c} nd {b,d} re two 3-connected intersecting seprting pirs on fce f in Ĝ nd let P be pth outside f from b to d. In prticulr, P does not pss through or c. As the pir b, d is 3-connected, it cnnot be seprted from the rest of the grph by ny other seprting pir. Let v be vertex tht gets seprted from b nd d when nd c re removed from the grph. Since v lies outside f, there is pth outside f from vi v to c. Since the grph is plnr, this pth must intersect P. Thus there is pth from v to b nd d tht does not pss through or c. This contrdicts the ssumption tht removl of nd c seprted v from b nd d. Definition 3.5 Cll set of vertices V V (Ĝ) seprble if there exists 3-connected seprting pir {,b} in V (Ĝ) such tht the removl of {,b} divides V into different connected components. Otherwise V is clled inseprble. Given n inseprble triple τ = {u,v,w}, define C τ = {x {u,v,w,x} is inseprble}. Note tht the nodes of simple cycle re trivilly inseprble becuse there re no 3- connected seprting pirs. The following lemm sttes tht except for cycles, ll biconnected grphs hve 3-connected seprting pirs nd hence the sets C τ defined bove re the 3- connected components of such grph. 5

6 Lemm 3.6 Let G be biconnected plnr grph. If G is not 3-connected nd not cycle then G hs 3-connected seprting pir. Proof. Let G be neither 3-connected nor cycle nd let,b be seprting pir of G. If nd b re 3-connected then we re done. So ssume tht nd b re not 3-connected. Let f be fce spnned by nd b. Then nd b re connected by two vertex-disjoint pths, sy P 1 nd P 2, which form the boundry of f, nd the removl of {,b} seprtes these two pths. Since G is not single cycle, it hs more fces prt from f. Therefore f shres some of its edges with nother fce, sy f. Consider the common boundry between f nd f. The endpoints of this boundry, sy {u,v} hve three vertex-disjoint pths between them, nd hence re 3-connected. Both u nd v lie on P 1 or both lie on P 2, since otherwise P 1 nd P 2 will not be seprted on the removl of {,b}. Without loss of generlity, ssume tht u,v P 1. Let P 1 = { = v 1,v 2,...,v k = b} nd consider ll 3-connected pirs {v i,v j } of vertices tht lie on P 1. Pick pir, sy {v i,v j }, tht is mximlly prt on P 1. We clim tht {v i,v j } is seprting pir: if not, there exists pth outside f from v i to v j for some i < i, or from v j to v i for some j > j. In the first cse, {v i,v j } is 3-connected pir tht is further prt thn {v i,v j }, in the second cse the sme holds for {v i,v j }. But this contrdicts the choice of {v i,v j }. Hence, with inseprble triples we cn compute triconnected components. If triple of vertices is inseprble, then it is prt of the sme triconnected component. For distinct τ 1,τ 2, the sets C τ1 nd C τ2 re either disjoint or identicl. This llows us to identify ny such C τ with the lexicogrphicl smllest τ 0 (considering the lbels of vertices in τ lexicogrphiclly sorted) such tht C τ = C τ0. This is the pproch of Algorithm 1 below. Algorithm 1 Algorithm to decompose grph into triconnected components. Input: Biconnected plnr grph G = (V, E). Output: The triconnected components of G. 1: fix plnr embedding Ĝ of G. 2: for ll fces f of Ĝ do 3: S f {{u,v} {u,v} is 3-connected seprting pir tht spns f} 4: S f G bs f the set of 3-connected seprting pirs 5: for ll {u,v} S do 6: if {u, v} E then output 3-bond for {u, v} 7: compute the set of ll inseprble triples τ 1,...,τ k 8: for i 1 to k do {compute 3-connected components} 9: if h < i τ i τ h is seprble set then 10: C i τ i {crete new 3-connected component using inseprble triple τ i } 11: for j i + 1 to k do 12: if τ i τ j is n inseprble set then C i C i τ j 13: output the induced subgrph on C i without edges corresponding to 3-bonds, including virtul edges {s S s C i } First, the lgorithm computes ll 3-connected seprting pirs in the set S. From these, we get ll the 3-bonds. The for-loop from line 8 on computes the 3-connected components C τ : In line 9, we serch for the first inseprble triple τ / {C τh 1 h < i} tht cn be seprted from ll previous ones. In lines 11 nd 12, we serch for ll τ j C τi. By Lemm 3.6, 6

7 it suffices to consider the pirs in S to check whether set is seprble or not. The set C i finlly equls C τi. In line 13 we compute the triconnected component induced by C τi. An exmple of decomposition is provided in Figure 2. Ech step in the lgorithm cn be implemented in log-spce. For instnce, combintoril embedding for plnr grphs cn be computed in log-spce [AM00]. Seprting pirs, inseprble triples nd the triconnected components cn be computed in log-spce, mking orcle queries to undirected rechbility [Rei05]. c f b e G 1 f b c b c e T G 1 b G 2 d c d d d c d Ĝ G 3 G 2 G 4 G 3 G 4 Figure 2: The decomposition of biconnected plnr grph Ĝ. Its triconnected components re G 1,...,G 4 nd the corresponding triconnected component tree is T. In Ĝ, the pirs {,b} nd {c, d} re 3-connected seprting pirs. The inseprble triples re {, b, c}, {b, c, d}, {,c,d}, {,b,d}, {,b,f}, nd {c,d,e}. Hence the triconnected components re the induced grphs G 1 on {,b,f}, G 2 on {,b,c,d}, nd G 4 on {c,d,e}. Since the 3-connected seprting pir {c,d} is connected by n edge in Ĝ, we lso get {c,d} s triple-bond G 3. The virtul edges corresponding to the 3-connected seprting pirs re drwn with dshed lines. The triconnected component tree. Construct grph T such tht its nodes correspond to triconnected components nd seprting pirs, see Figure 2. There is n edge between triconnected component node nd seprting pir node if the vertices of the seprting pir re contined in the triconnected component. Two triconnected component nodes or seprting pir nodes do not shre n edge. It is esy to see tht T is tree, referred to s the triconnected component tree of G. Conversely, given T, we define grph(t) = G, the grph which hs the triconnected component tree T. We list some properties of T. Lemm 3.7 The grph T defined bove hs the following properties: 1. T is tree nd ll the leves of T re triconnected components. 2. Ech pth in T is n lternting pth of seprting pirs nd triconnected components. Hence, pth between two leves lwys contins n odd number of nodes nd therefore T hs unique center node. 3. With n rbitrry seprting pir node s root, T hs odd depth. 4. A 3-bond is introduced s child of seprting pir only s n indictor tht the vertices of the seprting pir hve n edge between them in G. Hence 3-bond is 7

8 lwys lef node. In [HT73] it is k-bond, where k is the number of components formed by the removl of the seprting pir. Observe, k is the number of children of its prent seprting pir nd cn be computed esily. Proof. We only show the first clim. Suppose T hs cycle C. By definition, C is n lternting cycle of seprting pirs nd triconnected components, C = (p 1,c 1,p 2,c 2...,p r,c r,p 1 ). Remove ny seprting pir p i from C. Then the triconnected components c i 1 nd c i remin connected through the other elements of the cycle, contrdicting the ssumption tht p i seprtes them. 4 Cnoniztion of Biconnected Plnr Grphs In this section, we give log-spce lgorithm to cnonize biconnected plnr grphs. For this, we define n isomorphism ordering on triconnected component trees which is similr to tht of Lindell s tree isomorphism ordering. We first give brief overview of Lindell s lgorithm nd then describe our cnoniztion procedure. 4.1 Overview of Lindell s Algorithm for Tree Cnoniztion Lindell [Lin92] gve log-spce lgorithm for tree cnoniztion. The lgorithm is bsed on n order reltion on trees defined below. The order reltion hs the property tht two trees S nd T re isomorphic if nd only if S = T. Becuse of this property it is clled cnonicl order. Clerly, n lgorithm tht decides the order cn be used s n isomorphism test. Lindell showed how to extend such n lgorithm to compute cnon for tree in logspce. Let S nd T be two trees with root s nd t, respectively. The cnonicl order is defined S < T if: 1. S < T, or 2. S = T but #s < #t, where #s nd #t re the number of children of s nd t, respectively, or 3. S = T nd #s = #t = k, but (S 1,...,S k ) < (T 1,...,T k ) lexicogrphiclly, where it is inductively ssumed tht S 1... S k nd T 1... T k re the ordered subtrees of S nd T rooted t the k children of s nd t, respectively. The comprisons in steps 1 nd 2 cn be mde in log-spce. Lindell proved tht even the third step cn be performed in log-spce using two-pronged depth-first serch, nd crosscompring only child of S with child of T. This is briefly described below: Find the number of miniml sized children of s nd t. If these numbers re different then the tree with lrger number of miniml children is declred to be smller. If equlity is found then remember the miniml size nd check for the next size. This process is continued till n inequlity in the sizes is detected or ll the children of s nd t re exhusted. If s nd t hve the sme number of children of ech size then ssume tht the children of s nd t re prtitioned into size-clsses (referred to s blocks in [Lin92]) in the 8

9 incresing order of the the sizes of the subtrees rooted t them. Tht is, the k children of s nd t re prtitioned into groups, such tht the i-th group is of crdinlity k i nd the subtrees in the i-th group ll hve size N i, where N 1 < N 2 <. It follows tht i k i = k nd i k in i = n 1. Then compre the children in ech size-clss recursively s follows: Cse 1, k = 0. Hence s nd t hve no children. They re isomorphic s ll one-node trees re isomorphic. We conclude tht S = T. Cse 2, k = 1. Recursively consider the grnd-children of s nd t. No spce is needed for the recursive cll. Cse 3, k 2. For ech of the subtrees S j compute its order profile. The order profile consists of three counters, c <, c > nd c =. These counters indicte the number of subtrees in the size-clss of S j tht re respectively smller thn, greter thn, or equl to S j. The counters re computed by mking cross-comprisons. Note, tht isomorphic subtrees in the sme size-clss hve the sme order profile. Therefore, it suffices to check tht ech such order profile occurs the sme number of times in ech size-clss in S nd T. To perform this check, compre the different order profiles of every size clss in lexicogrphic order. The subtrees in the size-clss i of S nd T, which is currently being considered, with count c < = 0 form the first isomorphism clss. The size of this isomorphism clss is compred cross the trees by compring the vlues of the c = vribles. If these vlues mtch then both trees hve the sme number of miniml children. Note tht the lexicogrphicl next lrger order profile hs the current vlue of c < + c = s its vlue for the c < -counter. This wy, one cn loop through ll the order profiles. If difference in the order profiles of the subtrees of S nd T is found then the lexicogrphicl smller order profile defines the smller tree. The lst order profile considered is the one with c < +c = = k for the current counters. If this point is pssed without uncovering n inequlity then the trees must be isomorphic nd it follows tht S = T. Since i k in i n, the following recursion eqution for the spce complexity holds. For ech new size clss, the work-tpe llocted for the former computtions cn be reused. ( ) n S(n) = mx{s(n i ) + O(log k i )} mx{ S + O(log k i )}, i i where k i 2 for ll i. It is not hrd to see tht S(n) = O(log n). 4.2 Isomorphism Order of Triconnected Component Trees We describe now n isomorphism order procedure for two triconnected component trees S nd T, corresponding to two biconnected plnr grphs G nd H, respectively. We root S nd T t seprting pir nodes s = {,b} nd t = {,b }, respectively, which re chosen rbitrrily. Note, n isomorphism test cn esily run through ll posibilities of choosing these roots. The rooted trees re denoted s S {,b} nd T {,b }. They hve seprting pir nodes k i 9

10 t odd levels nd triconnected component nodes t even levels. Figure 3 shows two trees to be compred. Our isomorphism ordering procedure is more complex thn Lindell s lgorithm, becuse ech node of the tree is seprting pir or triconnected component. Thus, unlike in the cse of Lindell s lgorithm, two leves in triconnected component tree re not lwys isomorphic. In [DLN08], log-spce cnoniztion lgorithm for 3-connected plnr grphs is described. Note tht, n obvious wy to cnonize triconnected component tree would be to invoke the lgorithm of [DLN08] long with Lindell s lgorithm. However, this pproch does not work. See for exmple, Figure 1 on pge 3. In the esiest cse, the components to these leves re not of the sme size. We strt by defining the size of triconnected component tree. S {,b} s b T {,b } t b s 1 G 1... S G1 S Gk T H1 T Hk... G k S 1 S lk T 1 T lk s l1... s lk t 1 H 1 t l1 H k t lk Figure 3: Triconnected component trees. Definition 4.1 For triconnected component tree T, the size of n individul component node C of T is the number n C of nodes in C. Note tht the seprting pir nodes re counted in in every component where they occur. The size of the tree T, denoted by T, is the sum of the sizes of its component nodes. Note tht the size of T is t lest s lrge s the number of vertices in grph(t), the grph corresponding to the triconnected component tree T. We define the isomorphism order < T for S {,b} nd T {,b } by first compring their sizes, then the number of children of s nd t. These two steps re exctly the sme s in Lindell s lgorithm. If equlity is found in these two steps, then in the third step we mke recursive comprisons of the subtrees of S {,b} nd T {,b }. However, here it does not suffice to compre the order profiles of the subtrees in the different size clsses s in Lindell s lgorithm explined bove. We need further comprison step to ensure tht G nd H re indeed isomorphic. To see this ssume tht s nd t hve two children ech, G 1, G 2 nd H 1, H 2 such tht G 1 = H 1 nd G 2 = H2. Still we cnnot conclude tht G nd H re isomorphic becuse it is possible tht the isomorphism between G 1 nd H 1 mps to nd b to b, but the isomorphism between G 2 nd H 2 mps to b nd b to. Then these two isomorphisms cnnot be extended to n isomorphism between G nd H. For n exmple see Figure 1 of Pge 3. 10

11 To hndle this, we introduce the notion of n orienttion of seprting pir. A seprting pir gets n orienttion from subtrees rooted t its children. Also, every subtree rooted t triconnected component node gives n orienttion to the prent seprting pir. If the orienttion is consistent, then we define S {,b} = T T {,b } nd we will show tht G nd H re isomorphic in this cse. Isomorphism order of two subtrees rooted t triconnected components. We consider the isomorphism order of two subtrees S Gi nd T Hj rooted t triconnected component nodes G i nd H j, respectively. We distinguish the following cses. Cse 1, G i nd H j re of different types. G i nd H j cn be either 3-bonds or cycles or 3-connected components. If the types of G i nd H j re different, we immeditely detect n inequlity, s it suffices to check whether ech of them is cycle or 3-bond or neither of them. We define cnonicl order mong subtrees rooted t triconnected components in this scending order: 3-bond, cycle, 3-connected component, such tht e.g. S Gi < T T Hj if G i is 3-bond nd H j is cycle. Cse 2, G i nd H j re 3-bonds. In this cse, S Gi nd T Hj re leves, immeditely define S Gi = T T Hj. Clerly, G i = Hj s ll 3-bonds re isomorphic. Cse 3, G i nd H j re cycles or 3-connected components. We construct the cnons of G i nd H j nd compre them bit-by-bit. To cnonize cycle, we trverse it strting from the virtul edge tht corresponds to its prent, nd then trversing the entire cycle long the edges encountered. There re two possible trversls depending on which direction of the strting edge is chosen. Thus, cycle hs two possible cnons. To cnonize 3-connected component G i, we use the log-spce lgorithm from Dtt, Limye, nd Nimbhorkr [DLN08]. Besides G i, the lgorithm gets s input strting edge nd combintoril embedding ρ of G i. We lwys tke the virtul edge {,b} corresponding to G i s prent s the strting edge. Then there re two choices for the direction of this edge, (,b) or (b, ). Further, 3-connected grph hs two plnr combintoril embeddings [Whi33]. Hence, there re four possible wys to cnonize G i. We strt the cnoniztion of G i nd H j in ll the possible wys (two if they re cycles nd four if they re 3-connected components), nd compre these cnons bit-by-bit. Let C g nd C h be two cnons to be compred. The bse cse is tht G i nd H j re lef nodes nd therefore contin no further virtul edges. In this cse we use the lexicogrphic order between C g nd C h. If G i nd H j contin further virtul edges then these edges re specilly treted in the bitwise comprison of C g nd C h : 1. If virtul edge is trversed in the construction of one of the cnons C g or C h but not in the other, then we define the one without the virtul edge to be the smller cnon. 2. If C g nd C h encounter virtul edges {u,v} nd {u,v } corresponding to child of G i nd H j, respectively, we need to recursively compre the subtrees rooted t {u,v} nd {u,v }. If we find in the recursion tht one of the subtrees is smller thn the other, then the cnon with the smller subtree is defined to be the smller cnon. 3. If we find tht the subtrees rooted t {u,v} nd {u,v } re equl then we look t the orienttions given to {u,v} nd {u,v } by their children. This orienttion, clled the reference orienttion, is defined below. If one of the cnons trverses the virtul edge 11

12 in the direction of its reference orienttion but the other one not, then the one with the sme direction is defined to be the smller cnon. We eliminte the cnons which were found to be the lrger cnons in t lest one of the comprisons. In the end, the cnons tht re not eliminted re the minimum cnons. If we hve minimum cnons for both G i nd H j then we define S Gi = T T Hj. The construction of the cnons lso defines n isomorphism between the subgrphs described by S Gi nd T Hj, i.e. grph(s Gi ) = grph(t Hj ). For single triconnected component this follows from Dtt, Limye, nd Nimbhorkr [DLN08]. If the trees contin severl components, then our definition of S Gi = T T Hj gurntees tht we cn combine the isomorphisms of the components to n isomorphism between grph(s Gi ) nd grph(t Hj ). Finlly, we define the orienttion given to the prent seprting pir of G i nd H j s the direction in which the minimum cnon trverses this edge. If the minimum cnons re obtined for both choices of directions of the edge, we sy tht S Gi nd T Hj re symmetric bout their prent seprting pir, nd thus do not give n orienttion. This finishes the description of the order for the cse of subtrees rooted t triconnected components. Observe, tht we do not need to compre the sizes nd the degree of the root nodes of S Gi nd T Hj in n intermedite step, s it is done in Lindell s lgorithm for subtrees. Tht is, becuse the degree of the root node G i is encoded s the number of virtul edges in G i. The size of S Gi is checked by the length of the miniml cnons for G i nd when we compre the sizes of the children of the root node G i with those of H j. Isomorphism order of two subtrees rooted t seprting pirs. The first three steps of the isomorphism ordering re performed similr to tht of [Lin92] mintining the order profiles. Now we ssume tht the subtrees re prtitioned into isomorphism clsses. The dditionl step involves comprison of orienttions given by the corresponding isomorphism clsses defined s follows: Let (G 1,...,G k ) be the children of the root {,b} of S {,b}, nd (S G1,...,S Gk ) be the subtrees rooted t (G 1,...,G k ). Similrly let (H 1,...,H k ) be the children of the root {,b } of T {,b } nd (T H1,...,T Hk ) be the subtrees rooted t (H 1,...,H k ). We first order the subtrees, sy S G1 T T S Gk nd T H1 T T T Hk, nd verify tht S Gi = T T Hi for ll i. If we find n inequlity then the one with the smllest index i defines the order between S {,b} nd T {,b }. Now ssume tht S Gi = T T Hi for ll i. Inductively, the corresponding split components re isomorphic, i.e. grph(s Gi ) = grph(t Hi ) for ll i. The next comprison concerns the orienttion of {,b} nd {,b }. We lredy explined bove the orienttion given by ech of the S Gi s to {,b}. We define reference orienttion for the root nodes {,b} nd {,b } which is given by their children. This is done s follows. We prtition (S G1,...,S Gk ) into clsses of isomorphic subtrees, sy I 1 < T... < T I p for some p k, nd similr (T H1,...,T Hk ) into I 1 < T... < T I p. It follows tht I j nd I j contin the sme number of subtrees for every j. Consider the orienttion given to {,b} by n isomorphism clss I j : For ech isomorphism clss I j we compute n orienttion counter, which is pir O j = (c j,c j ), where c j is the number of subtrees of I j which give one orienttion, sy (,b), nd c j is the number of subtrees from I j which give the other orienttion, (b,). The lrger number decides the orienttion given to {,b}. If these numbers re equl, or if ech component 12

13 in this clss is symmetric bout {,b} then no orienttion is given to {,b} by this clss, nd the clss is sid to be symmetric bout {, b}. Note tht in n isomorphism clss, either ll or none of the components re symmetric bout the prent. The reference orienttion of {,b} is defined s the orienttion given to {,b} by the smllest non-symmetric isomorphism clss. If ll isomorphism clsses re symmetric bout {,b}, then we sy tht {,b} hs no reference orienttion. We order ll the orienttion counters O j = (c j,c j ) such tht the first component c j is the counter for the reference orienttion of {,b}. Let O j = (d j,d j ) be the corresponding orienttion counters for the isomorphism clsses I j. Now we compre the orienttion counters O j nd O j for j = 1,...,p. If they re ll pirwise equl, then the grphs G nd H re isomorphic nd we define S {,b} = T T {,b }. Otherwise, let j be the smllest index such tht O j O j. Then we define S {,b} < T T {,b } if O j is lexicogrphiclly smller thn O j, nd T {,b } < T S {,b} otherwise. This finishes the definition of the order. For n exmple, see Figure 1. The grphs G nd H hve the sme triconnected component trees but re not isomorphic. In S {,b}, the 3-bonds form one isomorphism clss I 1 nd the other two components form the second isomorphism clss I 2, s they ll re pirwise isomorphic. The non-isomorphism is detected by compring the directions given to the prent seprting pir. We hve p = 2 isomorphism clsses nd for the orienttion counters we hve O 1 = O 1 = (0,0), wheres O 2 = (2,0) nd O 2 = (1,1) nd hence O 2 is lexicogrphiclly smller thn O 2. Therefore we hve T {,b } < T S {,b}. Summry of the steps in the isomorphism order. The isomorphism order of two triconnected component trees S nd T rooted t seprting pirs s = {,b} nd t = {,b } is defined S {,b} < T T {,b } if: 1. S {,b} < T {,b } or 2. S {,b} = T {,b } but #s < #t or 3. S {,b} = T {,b }, #s = #t = k, but (S G1,...,S Gk ) < T (T H1,...,T Hk ) lexicogrphiclly, where we ssume tht S G1 T... T S Gk nd T H1 T... T T Hk re the ordered subtrees of S {,b} nd T {,b }, respectively. To compute the order between the subtrees S Gi nd T Hi we compre lexicogrphiclly the cnons of G i nd H i nd recursively the subtrees rooted t the children of G i nd H i. Note, tht these children re gin seprting pir nodes. 4. S {,b} = T {,b }, #s = #t = k, (S G1 T... T S Gk ) = T (T H1 T... T T Hk ), but (O 1,...,O p ) < (O 1,...,O p) lexicogrphiclly, where O j nd O j re the orienttion counters of the j th isomorphism clsses I j nd I j of ll the S G i s nd the T Hi s. We sy tht two triconnected component trees S e nd T e re equl ccording to the isomorphism order, denoted by S e = T T e, if neither S e < T T e nor T e < T S e holds. The following theorem sttes tht two trees re = T -equl, precisely when the underlying grphs re isomorphic. Theorem 4.2 The biconnected plnr grphs G nd H re isomorphic if nd only if there is choice of seprting pirs e,e in G nd H such tht S e = T T e when rooted t e nd e, respectively. 13

14 Proof. Assume tht S e = T T e. The rgument is n induction on the depth of the trees tht follows the inductive definition of the isomorphism order. The induction goes from depth d to d+2. If the grndchildren of seprting pirs, sy s nd t, re = T -equl up to step 4, then we compre the children of s nd t. If they re equl then we cn extend the = T -equlity to the seprting pirs s nd t. When subtrees re rooted t seprting pir nodes, the comprison describes n order on the subtrees which correspond to split components of the seprting pirs. The order describes n isomorphism mong the split components. When subtrees re rooted t triconnected component nodes, sy G i nd H j, the comprison sttes equlity if the components hve the sme cnon, i.e. re isomorphic. By the induction hypothesis we know tht the children rooted t virtul edges of G i nd H j re isomorphic. The equlity in the comprisons inductively describes n isomorphism between the vertices in the children of the root nodes. Hence, the isomorphism between the children t ny level cn be extended to n isomorphism between the corresponding subgrphs in G nd H nd therefore to G nd H itself. The reverse direction holds obviously s well. Nmely, if G nd H re isomorphic nd there is n isomorphism tht mps the seprting pir {,b} of G to the seprting pir {,b } of H, then the triconnected component trees S {,b} of G nd T {,b } of H rooted respectively t {,b} nd {,b } will clerly be equl. Hence, such n isomorphism mpps seprting pirs of G onto seprting pirs of H. This isomorphism describes permuttion on the split components of seprting pirs, which mens we hve permuttion on triconnected components, the children of the seprting pirs. By induction hypothesis, the children (t depth d + 2) of two such triconnected components re isomorphic nd equl ccording to = T. More formlly, one cn rgue inductively on the depth of S {,b} nd T {,b }. 4.3 Spce Complexity of the Isomorphism Order Algorithm We nlyse the spce complexity of the isomorphism order lgorithm. The first two steps of the isomorphism order lgorithm cn be computed in log-spce s in Lindell s lgorithm [Lin92]. We show tht steps 3 nd 4 cn lso be performed in log-spce. We use the lgorithm of Dtt, Limye, nd Nimbhorkr [DLN08] to cnonize triconnected component G i of size n Gi in spce O(log n Gi ). Compring two subtrees rooted t triconnected components. For this, we consider two subtrees S Gi nd T Hj with S Gi = T Hj = N rooted t triconnected component nodes G i nd H j, respectively. The cses tht G i nd H j re of different types or re both 3-bonds re esy to hndle. Assume now tht both re cycles or 3-connected components. Then we strt constructing nd compring ll the possible cnons of G i nd H j. We eliminte the lrger ones nd mke recursive comprisons whenever the cnons encounter virtul edges simultneously. We cn keep trck of the cnons, which re not eliminted, in constnt spce. Suppose we construct nd compre two cnons C g nd C h nd consider the moment when we encounter virtul edges {,b} nd {,b } in C g nd C h, respectively. Now we recursively compre the subtrees rooted t the seprting pir nodes {,b} nd {,b }. Note, tht we cnnot fford to store the entire work-tpe content. It suffices to store the informtion of the cnons which re not eliminted, 14

15 which cnons encountered the virtul edges corresponding to {,b} nd {,b }, nd the direction in which the virtul edges {,b} nd {,b } were encountered. This tkes ltogether O(1) spce. When recursive cll is completed, we look t the work-tpe nd compute the cnons C G nd C h. Therefore, recompute the prent seprting pir of the component, where the virtul edge {,b} is contined. With look on the bits stored on the work-tpe, we cn recompute the cnons C g nd C h. Recompute for them, where {,b} nd {,b } re encountered in the correct direction of the edges nd resume the computtion from tht point. Although we only need O(1) spce per recursion level, we cnnot gurntee yet, tht the implementtion of the lgorithm described so fr works in log-spce. The problem is, tht the subtrees where we go into recursion might be of size > N/2 nd in this cse the recursion depth cn get too lrge. To get round this problem, we check whether G i nd H j hve lrge child, before strting the construction nd comprison of their cnons. A lrge child is child which hs size > N/2. If we find lrge child of G i nd H j then we compre them priori nd store the result of their recursive comprison. Becuse G i nd H j cn hve t most one lrge child ech, this needs only O(1) dditionl bits. Now, whenever the virtul edges corresponding to the lrge children from S Gi nd T Hj re encountered simultneously in cnon of G i nd H j, the stored result cn be used, thus voiding recursive cll. Compring two subtrees rooted t seprting pirs. Consider two subtrees S {,b} nd T {,b } of size N, rooted t seprting pir nodes {,b} nd {,b }, respectively. We strt compring ll the subtrees S Gi nd T Hj of S {,b} nd T {,b }, respectively. These subtrees re rooted t triconnected components nd we cn use the implementtion described bove. Therefore, we store on the work-tpe the counters c <,c =,c >. If they turn out to be pirwise equl, we compute the orienttion counters O j nd O j of the isomorphism clsses I j nd I j, for ll j. The isomorphism clsses re computed vi the order profiles of the subtrees, s in Lindell s lgorithm. When we return from recursion, it is n esy tsk to find {,b} nd {,b } gin, since triconnected component hs unique prent, which lwys is seprting pir node. Since we hve the counters c <,c =,c > nd the orienttion counters on the work-tpe, we cn proceed with the next comprison. Let k j be the number of subtrees in I j. The counters c <,c =,c > nd the orienttion counters need ltogether t most O(log k j ) spce. From the orienttion counters we lso get the reference orienttion of {,b}. Let N j be the size of the subtrees in I j. Then we hve N j N/k j. This would led to log-spce implementtion s in Lindell s lgorithm except for the cse tht N j is lrge, i.e. N j > N/2. We hndle the cse of lrge children s bove: we recurse on lrge children priori nd store the result in O(1) bits. Then we process the other subtrees of S {,b} nd T {,b }. When we rech the size-clss of the lrge child, we know the reference orienttion, if ny. Now we use the stored result to compre the orienttions given by the lrge children to their respective prent, nd return the result ccordingly. As seen bove, while compring two trees of size N, the lgorithm uses no spce for mking recursive cll for subtree of size lrger thn N/2, nd it uses O(log k j ) spce if the subtrees re of size t most N/k j, where k j 2. Hence we get the sme recurrence for 15

16 the spce S(N) s Lindell: S(N) mx j ( N S k j ) + O(log k j ), where k j 2 for ll j. Thus S(N) = O(log N). Note tht the number n of nodes of G is in generl smller thn N, becuse the seprting pir nodes occur in ll components split off by this pir. But we certinly hve n < N O(n 2 ) [HT73]. This proves the following theorem. Theorem 4.3 The isomorphism order between two triconnected component trees of biconnected plnr grphs cn be computed in log-spce. 4.4 The Cnon of Biconnected Plnr Grph Once we know the ordering mong the subtrees, it is stright forwrd to output the cnon of the triconnected component tree T. We trverse T in the tree isomorphism order s in Lindell [Lin92], outputting the cnon of ech of the nodes long with virtul edges nd delimiters. Tht is, we output [ while going down subtree, nd ] while going up subtree. We need to choose seprting pir s root for the tree. Since there is no distinguished seprting pir, we simply cycle through ll of them. Since there re less thn n 2 mny seprting pirs, log-spce trnsducer cn cycle through ll of them nd cn determine the seprting pir which, when chosen s the root, leds to the lexicogrphiclly minimum cnon of S. We describe the cnoniztion procedure for fixed root, sy {,b}. The cnoniztion procedure hs two steps. In the first step we compute wht we cll cnonicl list for S {,b}. This is list of the edges of G, lso including virtul edges. In the second step we compute the finl cnon from the cnonicl list. Cnonicl list of subtree rooted t seprting pir. Consider subtree S {,b} rooted t the seprting pir node {,b}. We strt with computing the reference orienttion of {, b} nd output the edge in this direction. This cn be done by compring the children of the seprting pir node {, b} ccording to their isomorphism order with the help of the orcle. Then we recursively output the cnonicl lists of the subtrees of {, b} ccording to the incresing isomorphism order. Among isomorphic siblings, those which give the reference orienttion to the prent re considered before those which give the reverse orienttion. We denote this cnonicl list of edges l(s,,b). If the subtree rooted t {,b} does not give ny orienttion to {,b}, then tke tht orienttion for {,b}, in which it is encountered during the construction of the bove cnon of its prent. Assume now, the prent of S {,b} is triconnected component. In the symmetric cse, S {,b} does not give n orienttion of {,b} to its prent. Then tke the reference orienttion which is given to the prent of ll siblings. Cnonicl list of subtree rooted t triconnected component. Consider the subtree S Gi rooted t the triconnected component node G i. Let {,b} be the prent seprting pir of S Gi with reference orienttion (,b). If G i is 3-bond then output its cnonicl list l(g i,,b) s (,b). If G i is cycle then it hs unique cnonicl list with respect to the orienttion (,b), tht is l(g i,,b). 16

17 Now we consider the cse tht G i is 3-connected component. Then G i hs two possible cnons with respect to the orienttion (,b), one for ech of the two embeddings. Query the orcle for the embedding tht leds to the lexicogrphiclly smller cnonicl list nd output it s l(g i,,b). If we encounter virtul edge {c,d} during the construction, we determine its reference orienttion with the help of the orcle nd output it in this direction. If the children of the virtul edge do not give n orienttion, we output {c,d} in the direction in which it is encountered during the construction of the cnon for G i. Finlly, the children rooted t seprting pir node {c,d} re ordered with the cnonicl order procedure. We give now n exmple. Consider the cnonicl list l(s,,b) of edges for the tree S {,b} of Figure 3. Let s i be the edge connecting the vertices i with b i. We lso write for short l (S i,s i ) which is one of l(s i, i,b i ) or l(s i,b i, i ). The direction of s i is s described bove. l(s,,b) = [ (,b) l(s G1,,b)... l(s Gk,,b) ], where l(s G1,,b) = [ l(g 1,,b) l (S 1,s 1 )... l (S l1,s l1 ) ]. l(s Gk,,b) = [ l(g k,,b) l (S lk,s lk ) ] Cnon for the biconnected plnr grph. This list is now lmost the cnon, except tht the nmes of the nodes re still the ones they hve in G. Clerly, cnon must be independent of the originl nmes of the nodes. The finl cnon for S {,b} cn be obtined by log-spce trnsducer which relbels the vertices in the order of their first occurrence in this cnonicl list nd outputs the list using these new lbels. Note tht the cnonicl list of edges contins virtul edges s well, which re not prt of G. However, this is not problem s the virtul edges cn be distinguished from rel edges becuse of the presence of 3-bonds. To get the cnon for G, remove these virtul edges nd the delimiters [ nd ] in the cnon for S {,b}. This is sufficient, becuse we describe here bijective function f which trnsforms n utomorphism φ of S {,b} into n utomorphism f(φ) for G with {,b} fixed. We get the following result. Theorem 4.4 A biconnected plnr grph cn be cnonized in log-spce. 5 Cnoniztion of Plnr Grphs In this section, we give log-spce lgorithm for the cnoniztion of plnr grphs. The min prt is to show how to cnonize connected plnr grphs. Then, if given grph is not connected, we compute its connected components in log-spce nd cnonize ech of these components. The cnons of the connected components re output in lexicogrphicl incresing order. Hence, from now on we ssume tht the given plnr grph is connected. We decompose plnr grph into its biconnected components nd then construct tree on these biconnected components nd rticultion points. We refer to this tree s the biconnected component tree. We lso refer to the components s biconnected component nodes nd rticultion point nodes. This tree is unique nd cn be constructed in log-spce [ADK08]. Similr to triconnected component trees, we put copy of n rticultion point into ech of the components formed by the removl of. Thus, n rticultion point hs copy in ech of the biconnected components obtined by its removl. 17

18 Note tht nive pproch is to color ll copies of n rticultion point with prticulr color nd check the isomorphism of these coloured biconnected components seprtely. However, this pproch does not work s we do not know priori which rticultion points from the two grphs will be mpped to ech other. Also, using this method, we cn not ensure in log-spce tht ll the copies of n rticultion re mpped to the copies of nother rticultion point. In the discussion below, we refer to copy of n rticultion point in biconnected component B s n rticultion point in B. Although n rticultion point hs t most one copy in ech of the biconnected components, the corresponding triconnected component trees cn hve mny copies of the sme rticultion point (if it belongs to seprting pir in the biconnected component). Given plnr grph G, we root its biconnected component tree t n rticultion point. During the isomorphism ordering of two such trees S nd T, we cn fix the root of S rbitrrily nd mke n equlity test for ll choices of roots for T. As there re n rticultion points, log-spce trnsducer cn cycle through ll of them for the choice of the root for T. We stte some properties of rticultion points. Lemm 5.1 Let B be biconnected component in S nd T(B) be its triconnected component tree. Then the following holds. 1. S hs unique center, similr to triconnected component tree. 2. If n rticultion point of S ppers in seprting pir node s in T(B), then it ppers in ll the triconnected component nodes which re djcent to s in T(B). 3. If n rticultion point ppers in two nodes C nd D in T(B), it ppers in ll the nodes tht lie on the pth between C nd D in T(B). Hence, there is unique node A in T(B) tht contins which is nerest to the center of T(B). We cll A the triconnected component ssocited with. Thus, we cn uniquely ssocite ech rticultion point contined in B with triconnected component in T(B). 5.1 Isomorphism Order for Biconnected Component Trees The isomorphism order for biconnected component trees is defined in three steps tht correspond to the first three steps of the isomorphism order for triconnected component trees in Section 4.2 on pge 13. We mention the min differences in the isomorphism ordering for biconnected component trees from tht of triconnected component trees. 1. The biconnected component nodes re connected by rticultion point nodes. The resulting grph is tree similr to the tree of triconnected component nodes nd seprting pir nodes. For rticultion points, we do not need the notion of orienttion. Insted, we color the copy of the prent rticultion point in biconnected component with distinct color nd then the pirwise isomorphism mong the subtrees of S nd T cn be extended to the isomorphism between the corresponding plnr grphs G nd H in stright forwrd wy. 2. When we compre biconnected components B nd B, then we do not hve n obvious, uniquely defined edge s root for the corresponding triconnected component trees T(B) nd T(B ). The nive pproch would be to cycle through ll seprting pirs nd finlly 18