3-connected Planar Graph Isomorphism is in Log-space

3-connected Planar Graph Isomorphism is in Log-space Samir Datta 1 , Nutan Lima y e 2 , and Pra jakta Nim bhork ar 2 1 Chennai Mathematical Institute, Chennai 603 103, India. sdatta@cmi .ac.in 2 The Institute of Mathematical Sciences, Chennai 600 113, India. nutan,praj akta@imsc.res.i n Abstract. W e sho w th at the isomorphism of 3-connected planar graphs can b e decided in deter- ministic log-space. This impro ves the previously kno wn b ound UL ∩ coUL of [13]. 1 In tro duct ion The general graph isomorphism problem is a w ell studied problem in computer science. Giv en t w o graphs, it deals w ith find ing a bijection b et ween the sets of v ertices of these t w o g raph s, such that the adjacencies are preserv ed. T he pr ob lem is in NP , but it is not kno wn to b e complete for NP . In fac t, it is kn o wn that if it is complete for NP , th en th e p olynomial hierarch y co llapses to its second lev el. On the other hand, no p olynomial time algorithm is kno wn. F or general graph isomorph ism NL and PL hardn ess is known [14 ], wh ereas for tr ees, L and NC 1 hardness is kn o wn, dep ending on the enco ding of the inp u t [6]. In literature, many sp ecial ca ses of this general graph isomorphism problem hav e b een studied. In some cases lik e trees [8], [3], or graphs w ith coloured v ertices and b ound ed colour classes [9], NC algorithms are kno wn. W e are inte rested in the case where the graphs under consideration are p lanar graphs. In [15], W einberg presented an O ( n 2 ) algorithm for testing isomorphism of 3-connected planar graphs. Hop cr oft and T arjan [5] extended this for general planar graphs, impro ving th e time complexit y to O ( n log n ). Hop croft and W ong [4] further impro ve d it to giv e a linear time algorithm. Its parallel complexit y w as first considered by Miller and Reif [10 ] and Ramac hand ran and Reif [11]. They ga v e an upp er b ound of A C 1 . Recen tly Thierauf and W agner [13] improv ed it to UL ∩ coUL for 3-co nn ected planar graph s. They also p ro v ed that this pr oblem is hard for L . In this pap er, we give a log-space alg orithm for 3-connecte d planar grap h isomorp hism, thereb y settling its complexit y . Thierauf and W agner u s e shortest paths b et we en no d es of a graph to obtain a canonical spanning tree. A s y s tematic trav ersal of this tree generates a canonical form for th e graph . T he b est kno wn u pp er b ound for shortest paths in p lanar grap h s is UL ∩ coUL [13]. Thus the total complexit y of their algorithm go es to UL ∩ coUL , despite the fact that all other steps can b e done in L . W e identify that their algorithm hinges on making a systematic tra versal of the graph in canonical w a y . Thus w e b ypass the s tep of fin ding sh ortest paths and give an orthogonal approac h for fi nding su c h a tra v ersal. W e use the notion of unive rsal exploration sequences (UXS) defined in [7]. Giv en a graph on n v ertices with maximum degree d , a UXS is nothing but a p olynomial length strin g ov er { 0 , . . . , d − 1 } . Such a sequence can b e used to tra v erse the graph for a chosen com bin atorial embed ding ρ , starting vertex u and a starting ed ge e = { u, v } . Reingold [12] prov ed th at suc h a u niv ersal sequence can b e constructed in L . Using this result, w e canonize a 3-connected planar graph in log- space. T o our kno wledge, this is the b est u pp er b ound for th is class of graph s. In Section 2, w e giv e some basic defin itions of complexit y classes and formally define the notion of univ ersal exploration sequences. In Section 3, w e describ e our lo g-space algorithm. W e conclude with a discussion of op en problems in S ection 4. 2 Preliminaries In this sectio n, we giv e a brief introdu ction of the graph isomorph ism problem an d the n otion of unive rsal exploration sequences. 2.1 Univ ersal E xploration Sequences Let G = ( V , E ) b e a d -regular grap h , with giv en com b inatorial emb ed ding ρ . T he edges aroun d an y v ertex u can b e num b ered { 0 , 1 , . . . , d − 1 } according to ρ arbitrarily in clo ckwise order. A sequence τ 1 τ 2 . . . τ k ∈ { 0 , 1 , . . . , d − 1 } k and a s tarting edge e 0 = ( v − 1 , v 0 ) ∈ E , define a w alk v − 1 , v 0 , . . . v k as follo ws: F or 0 ≤ i ≤ k , if ( v i − 1 , v i ) is the s th edge of v i , let e i = ( v i , v i +1 ) b e ( s + τ i ) th edge of v i mo dulo d . Definition 1. Uni versal Explor ation se que nc es (UX S): A se quenc e τ 1 τ 2 . . . τ l ∈ { 0 , 1 , . . . d − 1 } l is a universal explor ation se quenc e for d - r e gular gr aphs of size at most n if for every c onne cte d d -r e gular gr aph on at most n vertic es, any numb ering of its e dges, and any starting e dge, the walk obtaine d v isits al l the ve rtic es of the gr aph. F ollo wing lemma suggests that UXS can b e constru cted in L [12]: Lemma 1. Ther e exists a l o g- sp ac e algorithm th at takes as input (1 n , 1 D ) and pr o duc es an ( n, D ) -universal explor ation se quenc e. 2.2 The Graph Isomorphism Problem Definition 2. Gr aph isomorphism: Two gr aphs G 1 = ( V 1 , E 1 ) and G 2 = ( V 2 , E 2 ) ar e said to b e isomorph ic if ther e is a bije ction φ : V 1 → V 2 such that ( u, v ) ∈ E 1 if and only if ( φ ( u ) , φ ( v )) ∈ E 2 . Let GI b e th e problem of fin ding s u c h a b ijection φ giv en t wo graph s G 1 , G 2 . Let Planar-GI b e the sp ecial case of GI when the giv en grap h s are planar. 3-connected p lanar graph isomorphism problem is a sp ecial case of Planar-GI when the graphs are 3-connected planar graphs. W e recall the definition of 3-connected planar graphs here: A graph G is connected if there is a path b et we en an y t w o vertic es in G. A verte x v ∈ V is an articulation p oint if G − v is not connected. A pair of v ertices u, v ∈ V is a separation pair if G ( V \ { u, v } ) is not conn ected. A biconnected graph con tains no articulatio n p oin ts. A 3-connected graph con tains no separation pairs. 3 Log-space Algorithm for 3-connected Planar-GI In this section, w e prov e follo wing theorem: Theorem 1. Given two 3 -c onne cte d planar gr aphs G and H , de ciding whether G i s isomorphic to H c an b e done in L . F or general planar grap h s, the b est kn o wn p arallel algorithm runs in AC 1 [10]. Th ierauf and W agner [13] r ecen tly impro ved the b oun d for the case of 3-connected planar graphs to UL ∩ coUL . This case is easier due to a result b y Whitney [16] that ev ery planar 3-connected graph h as pr ecisely t wo embedd ings on a sp here, wh ere one em b edd ing is the mirror image of the other. Moreo ve r, one can efficien tly compu te these embedd ings. Using these emb eddings, Thierauf and W agner compute a co de for a graph, such that iso- morphic graphs w ill ha ve the s ame co de. A code with this prop erty is cal led a canonical co de 2 for the grap h . They construct it via a spann in g tree, which dep ends up on the planar em b ed- ding of the graph. Bourk e, T ew ari and Vino d c handran [2] pro ve d th at planar r eac h ab ility is in UL ∩ coUL . Th ierauf and W agner extend their result for computin g distances in planar graph s in UL ∩ coUL and crucially u se this in th e co nstr uction of the s panning tree. O nce this sp anning tree is constru cted, a canonical cod e can b e obtained in L . Our appr oac h b ypasses the sp anning tree constru ction step and thus eliminates distance computations. In that sense, we b elieve that this is a completely new approac h for computing canonical co des for 3-connected planar graphs. Our algorithm can b e outlined as follo ws: 1. Giv en a 3-connected planar graph G = ( V , E ), find its planar emb ed ding ρ . 2. Mak e the graph 3-regular ca nonically for this em b edd ing ρ to obtain an edge-coloured graph G ′ as describ ed in algorithm 3.1. 3. Find the canon of G ′ using algorithm 3.2. The step 1 is in log-space due to a result by Allender and Maha jan [1]. W e pro ve that steps 2 and 3 can also b e d one in log-space. S tep 3 uses the idea of UXS introd uced by Koucky []. S tep 2 essen tially do es the prep ro cessing in order to mak e step 3 applicable. The canonical co d e th us constru cted is sp ecific to th e c hoice of th e com binatorial em b edding, the starting edge, and the s tarting v ertex. Given t w o grap h s G and H , we fix th ese arbitrarily for G and cycle through b oth emb eddings and all c hoices of the s tarting edge and the starting v ertex for H , comparing th e co des for ea ch of them. As th er e are only p olynomially man y c hoices, this lo op run s in L . 3.1 Making the graph 3-regular In this section, w e describ e the p r o cedure to mak e the graph 3-regular. In Sectio n 3.2, w e use Reingold’s construction for UXS [12] to come u p with a canonical co de. As Reingold’s construction [12] for UXS requ ir es the graph to ha v e constan t degree, w e do this p repro cessing step. In Lemma 2, we pro ve that t wo giv en graph s are isomorp hic if and only if they are isomorphic after the p repro cessing step. W e n ote th at after the prepr o cessing step, the graph do es not remain 3-connected, h o w ev er, th e em b edding of the new graph is inh er ited fr om the giv en graph. Hence even the new graph has only t wo p ossible embed d ings. W e n o w d escrib e the prepro cessing steps in Algorithm 3.1. Note that the new graph th u s obtained has 2 | E | v ertices. Algorithm 1 Pro cedur e to get a 3-regular p lanar graph G ′ from 3-connected planar graph G . Input: A 3-connected planar graph G wi th p lanar combinatori al embedd ing ρ . Output: A 3-regular planar graph G ′ on 2 m vertices, with edges coloured 1 and 2 and planar com binatorial em b edd ing ρ ′ . for all v i ∈ V do Replace v i of b y a cycle { v i 1 , . . . , v id i } on d i vertices , where d i is the degree of v i . The d i edges { e i 1 , . . . , e id i } incident to v i in G are no w incident t o { v i 1 , . . . , v id i } respectively . Colour the cycle edges with colo ur 1. Colour e i 1 , . . . , e id i by colour 2. end for Lemma 2. Given tw o 3 -c onne cte d pl anar gr aphs G 1 , G 2 , G 1 ∼ = G 2 if and only if G ′ 1 ∼ = G ′ 2 wher e the isomorphism b e twe en G ′ 1 and G ′ 2 r esp e cts c olours of the e dges. 3 Pr o of. Let G 1 = ( V 1 , E 1 ) and G 2 = ( V 2 , E 2 ) b e t wo 3-connected planar graphs with p lanar com b inatorial embed d ings ρ 1 and ρ 2 resp ectiv ely . Let φ : V 1 → V 2 b e an isomorphism b et ween the oriented graphs ( G 1 , ρ 1 ) and ( G 2 , ρ 2 ). By isomorphism of orien ted grap h s w e mean that the graphs are isomorph ic for the fixed emb eddings, in our case ρ 1 and ρ 2 . Constru ct G ′ 1 and G ′ 2 preserving the orienta tion of original edges from G 1 and G 2 resp ectiv ely . Let the orienta tions b e ρ ′ 1 and ρ ′ 2 . By our construction, edges aroun d a vertex in G 1 (resp ectiv ely G 2 ) get the same com b inatorial em b edding around the corresp ond ing cycle in G ′ 1 ( G ′ 2 ). Consider an edge { v i , v j } in E 1 . Let φ ( v i ) = u k and φ ( v j ) = u l . { u k , u l } ∈ E 2 . Let corresp ondin g edge in G ′ 1 b e { v i p , v i q } and that in G ′ 2 b e { u k r , u k s } . Th en defin e a map φ ′ : V ′ 1 → V ′ 2 suc h that φ ′ ( v i p ) = u k r and φ ′ ( v j q ) = u k s . It is easy to see that φ ′ is an isomorphism for edge-col oured orien ted graphs ( G ′ 1 , ρ ′ 1 ) and ( G ′ 2 , ρ ′ 2 ). No w let φ ′ b e an isomorphism b etw een oriented graphs ( G ′ 1 , ρ ′ 1 ) and ( G ′ 2 , ρ ′ 2 ). Let e = { v i p , v i q } ∈ E ′ 1 where v i p and v i q corresp ond to the same v ertex v i in G 1 . Then colour ( e ) = 1 and e ′ = { φ ′ ( v i p ) , φ ′ ( v i q ) } ∈ E ′ 2 and col our ( e ′ ) = 1. Thus φ ′ maps copies of the same ve rtex of G 1 to copies of a single vertex of G 2 . Hence a map φ can b e deriv ed from φ ′ in a n atural wa y . It is ea sy to see that φ is an isomorphism b et ween oriented graph s ( G 1 , ρ 1 ) and ( G 2 , ρ 2 ). 3.2 Obtaining t he canonical co de Lemma 2 suggests th at for giv en em b edd ings ρ 1 , ρ 2 of G 1 and G 2 , it suffices to c hec k the 3-regular orien ted graphs ( G ′ 1 , ρ ′ 1 ) and ( G ′ 2 , ρ ′ 2 ) for isomorphism. This can b e done as follo ws: Algorithm 2 Pro cedur e canon ( G, ρ, v , e = ( u, v )) Input: Edge-coloured graph G = ( V , E ) with maxim um degree 3 and com binatorial emb edding ρ , starting vertex v , starting edge e = ( u, v ) Output: canon of G . Construct a ( n, 3) unive rsal exploration sequence U . With starti ng vertex v ∈ V and edge e = ( u, v ) inciden t to it, trav erse G according to U and ρ and output the labels of the vertices. Give lab els to the vertices acco rding to their first o ccu rrence in this output sequence. F or ev ery ( i, j ) in this labelling, output w heth er ( i, j ) is an ed ge or not. If it is an edge, out put its colour. This giv es a canon for the graph. Lemma 3. L et σ 1 = canon ( G ′ 1 , ρ ′ 1 , v 1 , e 1 = ( u 1 , v 1 )) and σ 2 = canon ( G ′ 2 , ρ ′ 2 , v 2 , e 2 = ( u 2 , v 2 )) . If σ 1 = σ 2 then G ′ 1 ∼ = G ′ 2 . F urther, if G ′ 1 ∼ = G ′ 2 then for some choic e of ρ ′ 2 , v 2 , e 2 , σ 1 = σ 2 . Pr o of. If G ′ 1 ∼ = G ′ 2 , then there is a bijectio n φ : V ′ 1 → V ′ 2 for corresp onding em b eddings ρ ′ 1 , ρ ′ 2 . Let e 1 = ( u, v ) ∈ E ′ 1 . Th en e 2 = ( φ ( u ) , φ ( v )) ∈ E ′ 2 . Let e 1 and e 2 b e c hosen as sta rting edges and v and φ ( v ) as starting vertice s for trav ersal usin g UXS U f or ( G ′ 1 , ρ ′ 1 ) and ( G ′ 2 , ρ ′ 2 ) resp ectiv ely . Let T 1 and T 2 b e the output sequen ce. If a v ertex w ∈ V ′ 1 o ccurs at p osition l in T 1 then φ ( w ) ∈ V ′ 2 o ccurs at p osition l in T 2 . Thus the sequences are canonical when pro j ected d o wn to the first o ccurrences and hence σ 1 = σ 2 . Let σ 1 = σ 2 = σ . The lab els of ve rtices in σ are j u st a relabelling of vertic es of V ′ 1 and V ′ 2 . These relab ellings are some p ermutat ions, s a y π 1 and π 2 . Then π 1 · π − 1 2 : V ′ 1 → V ′ 2 is a bijection. After constructing canonical co de σ ′ for a graph G ′ , it remains to construct canonical co de σ for the original graph G . F or eac h edge ( i, j ) of colour 2 in σ ′ , trav erse along the edges coloured 1 starting from i and find the minimum among the v ertices visited. Let it b e p . Rep eat the pro cess for j . Let the minim um v ertex visited along edges of colour 1 b e q . Output th e edge 4 ( p, q ). The sequen ce th u s obtained con tains n lab els for v ertices, eac h b et wee n { 1 , 2 , . . . , 2 m } . This can further b e con v erted in to a sequence w ith lab els for v ertices b et wee n { 1 , 2 , . . . , n } b y finding the rank of eac h of the lab els. T his giv es us σ . Correctness follo ws from the fact th at v ertices connected with edges of colour 1 are copies of the same v ertex in G , hence th ey should get the s ame n umb er . Clearly , eac h of the ab o v e steps can b e p erformed in L and hence the algo rithm runs in L . This pro ves Theorem 1. 4 Conclusion Our note settles the op en qu estion ment ioned in [13] b y giving a log-space algorithm for 3- connected planar graph isomorph ism. The most c hallenging question is to settle the complexity of the general graph isomorphism problem. T he other imp ortant goal is to impro ve up on the A C 1 upp er b oun d of [10] for planar graph isomorphism. 5 Ac kno wledgmen t W e thank Jacob o T oran, V. Arvind, and Meena Maha jan for helpful discussions. References 1. Eric Allend er and Meena Maha jan. The complexity of planarity testing. In ST ACS ’ 00: Pr o c e e dings of the 17th Annual Symp osium on The or etic al Asp e cts of Computer Scienc e , pages 87–98, 2000. 2. 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