Fast Isomorphism Testing of Graphs with Regularly-Connected Components

F ast Isomorphism T es ting of Graphs with Reg ularly-Connected Comp onen ts Jos ´ e Luis L´ op ez-Presa DIA TEL, Univ ersidad P olit´ ecnica de Madrid Madrid, Spain jllop ez@diatel.upm.es An tonio F ern´ andez An ta Institute IMDEA Net works Madrid, Spain an tonio.f ernandez@imdea.org Abstract The Graph Iso morphism problem has b oth theoretical and pra ctical interest. In this pa- per we present an algorithm, called c onauto-1.2 , that eﬃciently tests whether t wo graphs are isomorphic, a nd ﬁnds an is omorphism if they are. This algo rithm is a n improved version of the algor ithm c onauto , which ha s b een shown to b e very fast for random graphs a nd several families of har d g raphs [9]. In this pap er we establish a new theorem that allows, at v er y lo w cost, the easy disco very of many automorphisms. This result is esp ecially suited for g raphs with regular ly connected c o mpo nents, and can b e applied in any isomo rphism testing and canonica l lab eling alg o rithm to drastically improv e its p erfor mance. In particular, algorithm conauto -1.2 is obtained by t he application of this result to conauto. The re sulting algorithm preserves all the nice fea tur es of conauto, but drastically improves the testing of graphs with regular ly connected comp onents. W e run extensiv e exper iment s, which sho w that the most popular algorithms (namely , naut y [10, 11] and bliss [8]) ca n not comp ete with conauto-1.2 fo r these gra ph families. 1 1 In tro d uction The Graph Isomorphism problem (GI) is of b oth theoretical and practical in terest. GI tests w hether there is a one-to-one mapp ing b et we en the vertic es of t wo graphs that preserve s the arcs. This problem has applications in man y ﬁelds, like pattern recognition and computer vision [3], data mining [17], VLSI lay out v alidation [1], and c h emistry [5, 15]. At the theoretical lev el, its main theoretical inte r est is that it is n ot kno wn wh ether GI is in P or whether it is NP-complete. Related W ork It w ould b e nice to ﬁn d a complete graph-inv arian t 1 computable in p olynomial time, wh at w ould allo w testing graphs for isomorph ism in p olynomial time. Ho we ver, n o suc h in v ariant is kn o wn, and it is unlikely to exist. Note, though, that there are many simple instances of GI, and that many families of graph s can b e tested for isomorphism in p olynomial time: trees [2], planar graphs [7 ], graphs of b ounded degree [6], etc. F or a review of the theoretical results related to GI see [9, 13]. The m ost inte r esting p ractical app roac hes to th e GI problem are (1) th e direct approac h, wh ich uses bac ktr acking to ﬁnd a matc h b et we en the graph s, using tec h niques to p rune th e searc h tree, and (2) computing a c ertiﬁc ate 2 of eac h of the graph s to test, and then compare the certiﬁcates directly . The direct appr oac h can b e u sed for b oth graph and subgraph isomorp hism (e.g. vf2 [4] and Ullman’s [16] algorithms), bu t has p roblems when dealing w ith highly regular graphs with a relativ ely small automorphism group. In this case, ev en th e use of heuristics to prun e the searc h space frequently do es not prev ent the prop osed algorithms f rom exp loring paths equiv alen t to those already tested. T o a voi d this, it is necessary to ke ep trac k of discov ered automorphisms , and use this information to aggressiv ely prun e the searc h space. On the other hand, us in g certiﬁcates, since t w o isomorph ic graphs ha ve the same canonical lab eling, their certiﬁcates can b e compared directly . This is the approac h used by the w ell-known algorithm nauty [10, 11], and th e algorithm bliss [8] (whic h has b etter p er f ormance than nauty for some graph families). This approac h requires computing the fu ll automorphism group of the graph (at least a s et of generators). In most cases, these algorithms are faster than th e on es that use the direct approac h . Algorithm c onauto [9 ] uses a n ew approac h to graph isomo rphism 3 . It co mbines the use of disco ve r ed automorphisms with a bac ktrac kin g algorithm that tries to ﬁnd a matc h of the graphs without th e need of generating a canonical f orm. T o test graphs of n no des conauto u s es O ( n 2 log n ) bits of memory . Ad ditionally , it ru ns in p olynomial time (on n ) with high probabilit y for random graphs. In real exp eriments, for seve r al families of inte r esting hard graphs, conauto is faster than naut y an d vf 2, as shown in [9]. F or example Miy azaki’s graph s [12], are v ery hard for vf2, naut y , and b liss, but conauto handles them eﬃciently . Ho w eve r , it w as found in [9] that some families of graphs bu ilt f r om regularly connected comp onen ts (in particular, fr om strongly regular graphs) are not h an d led eﬃcien tly by any of the algorithms ev aluate d. Wh ile conauto runs fast wh en the tested graphs are isomorph ic, it is v ery slow w hen the graphs are not isomorphic. Con tribut ions In this p ap er we establish a new th eorem that all o ws, at very lo w cost, the easy d isco v ery of man y automorphisms. Th is result is esp ecially s uited for graphs with regularly 1 A complete graph-inv arian t is a function on a graph that gives the same result for isomorphic graphs, and diﬀeren t results for non-isomorphic graphs. 2 A certiﬁcate of a graph is a canonical labeling of the graph. 3 A preliminary version of conauto has been in clu d ed in the LEDA C++ class library of algorithms [14]. 2 connected comp onen ts, and can b e applied in any direct isomorphism testing or canonical lab eling algorithm to dr astically impr ov e its p erformance. Then, a new algorithm, called c onauto -1.2 , is prop osed. This algorithm is obtained by impro vin g conauto with tec hniques deriv ed from the ab o ve mentio n ed theorem. In p articular, conauto-1.2 reduces th e b ac ktrac king n eeded to explore ev ery plausible path in the searc h space with r esp ect to conauto. The resulting algorithm p reserv es all the nice features of conauto, bu t drastically improv es the testing of some graphs, lik e those w ith regularly connected comp onents. W e ha ve carried out exp eriments to compare th e pr actical p erform ance of conauto-1.2, naut y , and bliss, with diﬀerent families of graph s bu ilt by regularly connecting copies of small comp onent s . The exp eriments show that, for this type of constru ction, conauto-1.2 n ot only is the fastest, b ut also has a v ery regular b eha vior. Structure In Section 2, we d eﬁ ne the basic theoretical concepts used in algorithm conauto- 1.2 an d present the theorems on whic h its correction relies. Next, in Section 3 w e describ e the algorithm itself. Then, S ection 4 describ es the graph families used for th e tests, and sh o w the practical p erformance of conauto-1.2 compared with conauto, naut y and bliss f or these families. Finally w e put forwa r d our conclusions and prop ose new w ays to imp ro ve conauto-1.2. 2 Theoretical F oundation 2.1 Basic Deﬁnitions A dir e cte d g r aph G = ( V , R ) consists of a ﬁ nite non -emp t y set V of vertice s and a bin ary r elation R , i.e. a su bset R ⊆ V × V . The elements of R are called ar cs . An arc ( u, v ) ∈ R is considered to b e oriente d from u to v . An undir e cte d gr aph is a graph w hose arc set R is symmetrical, i.e. ( u, v ) ∈ R iﬀ ( v , u ) ∈ R . F rom no w on, w e w ill use the term gr aph to refer to a dir e cte d gr aph . Deﬁnition 1 An isomorphism of gr aphs G = ( V G , R G ) and H = ( V H , R H ) i s a b ije ction b etwe en the vertex sets of G and H , f : V G − → V H , such that ( v , u ) ∈ R G ⇐ ⇒ ( f ( v ) , f ( u )) ∈ R H . Gr aphs G and H ar e c al le d isomorphic , written G ≃ H , if ther e i s at le ast one isomorphism of them. A n automorphism of G is an isomorphism of G and itself. Giv en a graph G = ( V , R ), R can b e represen ted by an adjac ency matrix A dj ( G ) = A with size | V | × | V | in the follo win g w ay: A uv =  0 if ( u, v ) / ∈ R ∧ ( v , u ) / ∈ R 1 if ( u, v ) / ∈ R ∧ ( v , u ) ∈ R 2 if ( u, v ) ∈ R ∧ ( v , u ) / ∈ R 3 if ( u, v ) ∈ R ∧ ( v , u ) ∈ R Let G = ( V , R ) b e a graph, and A dj ( G ) = A its adjacency matrix. Let V 1 ⊆ V and v ∈ V , the available de gr e e of v in V 1 under G , denoted b y ADe g ( v , V 1 , G ), is th e d egree of v with resp ect to V 1 , i.e., the 3-tuple ( D 3 , D 2 , D 1 ) where D i = |{ u ∈ V 1 : A vu = i }| for i ∈ { 1 , 2 , 3 } . The p redicate HasLinks ( v , V 1 , G ) says if v has an y n eigh b or in V 1 , i.e., A D e g ( v, V 1 , G ) 6 = (0 , 0 , 0). Extendin g the notation, let V 1 , V 2 ⊆ V ; if ∀ u, v ∈ V 1 , ADe g ( u, V 2 , G ) = ADe g ( v , V 2 , G ) = d , then, we d enote ADe g ( V 1 , V 2 , G ) = d . HasLinks ( V 1 , V 2 , G ) is d eﬁned similarly . W e will say a 3-tuple ( D 3 , D 2 , D 1 ) ≺ ( E 3 , E 2 , E 1 ) when the ﬁrs t one precedes the second one in lexicographic order. This notation w ill b e used to order the a v ailable degrees of v ertices and sets. 3 2.2 Sp eciﬁc Not at ion and Deﬁnitions for the A lgor ithms It will b e necessary to in tro d uce some sp eciﬁc notation to b e u sed in the sp eciﬁcation of ou r algorithms. Lik e other isomorp hism testing algorithms, ours relies on vertex classiﬁcation. Let us start deﬁning what a partition is, and the partition concatenati on op eration. A p artition of a set S is a sequence S = ( S 1 , ..., S r ) of disjoin t nonempty subsets of S such th at S = S r i =1 S i . The sets S i are called the c el ls of S . The empty partition will b e d enoted by ∅ . Deﬁnition 2 L et S = ( S 1 , ..., S r ) and T = ( T 1 , ..., T s ) b e p artitions of two disjoint sets S and T , r esp e ctively. The concatenation of S and T , denote d S ◦ T , is the p artition ( S 1 , ..., S r , T 1 , ..., T s ) . Cle arly, ∅ ◦ S = S = S ◦ ∅ . Let G = ( V , R ) b e a graph , v ∈ V , V 1 ⊆ V \ { v } . The vertex p artition of V 1 b y v , denoted PartitionByV ertex ( V 1 , v , G ), is a partition ( S 1 , ..., S r ) of V 1 suc h that for all i, j ∈ { 1 , ..., r } , i > j implies AD e g ( S i , { v } , G ) ≺ ADe g ( S j , { v } , G ). Let V 1 , V 2 ⊆ V . The set p art ition of V 1 b y V 2 , denoted P artitionBySet ( V 1 , V 2 , G ), is a partition ( S 1 , ..., S r ) of V 1 suc h that for all i, j ∈ { 1 , ..., r } , i > j imp lies ADe g ( S i , V 2 , G ) ≺ ADe g ( S j , V 2 , G ). Deﬁnition 3 L et G = ( V , R ) b e a gr aph, and S = ( S 1 , ..., S r ) a p artition of V . L et v ∈ S x for some x ∈ { 1 , ..., r } . The ve r tex reﬁnement of S by v , denote d V ertexR eﬁnement ( S , v , G ) is the p artition T = T 1 ◦ ... ◦ T r such that for al l i ∈ { 1 , ..., r } , T i is the empty p artition ∅ if ¬ HasLinks ( S i , V , G ) , and PartitionByV ertex ( S i \ { v } , v , G ) otherwise. S x is the p iv ot set and v is the piv ot v ertex . Deﬁnition 4 L et G = ( V , R ) b e a gr aph, and S = ( S 1 , ..., S r ) a p artition of V . L et P = S x for some x ∈ { 1 , ..., r } b e a giv en piv ot set . The set reﬁn ement of S by P , denote d SetR eﬁnement ( S , P , G ) is the p artition T = T 1 ◦ ... ◦ T r such that for al l i ∈ { 1 , ..., r } , T i is the empty p artition ∅ if ¬ HasLinks ( S i , V , G ) , and PartitionBySet ( S i , P , G ) otherwise. Once w e h a ve pr esented the p ossible partition reﬁnements that ma y b e applied to partitions, w e can build sequences of partitions in wh ic h an initial partition (for examp le the one with one cell con taining all the vertic es of a graph) is iterativ ely reﬁned usin g th e tw o p reviously deﬁned reﬁnements. V ertex reﬁnemen ts are tagged as VER TEX (if th e pivot set has only one vertex), SET (if a set reﬁn emen t is p ossible with some piv ot set), or BA CKTR ACK (when a vertex reﬁnement is p erformed with a pivot set w ith m ore than one v er tex). Deﬁnition 5 L et G = ( V , R ) b e a gr aph. A sequence of partitions for g r aph G is a tuple ( S , R , P ) , wher e S = ( S 0 , ..., S t ) , ar e the p artitio ns themselves, R = ( R 0 , ..., R t − 1 ) indic ate the typ e of r eﬁne- ment applie d at e ach step, and P = ( P 0 , ..., P t − 1 ) cho ose the pivot set use d for e ach r eﬁnement step, such that al l the fol lowing statements hold: 1. F or al l i ∈ { 0 , ..., t − 1 } , R i ∈ { VER TEX , SET , BACKTRA CK } , and P i ∈ { 1 , ..., |S i |} . 2. F or al l i ∈ { 1 , ..., t − 1 } , let S i = ( S i 1 , ..., S i r i ) , V i = S r i j =1 S i j . Then: (a) R i = SET implies S i +1 = SetR eﬁnement ( S i , S i P i , G ) . (b) R i 6 = SET implies S i +1 = V ertexR eﬁnement ( S i , v , G ) for some v ∈ S i P i . 3. L et S t = ( S t 1 , ..., S t r ) , V t = S r j =1 S t j , then for al l S t x ∈ S t , | S t x | = 1 or ¬ H asLinks ( S t x , V t , G ) . 4 F or conv enience, for all l ∈ { 1 , ..., t − 1 } , by level l we refer to the tuple ( S l , R l , P l ) in a sequence of partitions. Lev el t is ident iﬁ ed b y S t , since R t and P t are not deﬁned. W e will now in tro d uce the concept of compatibilit y among partitions, and then deﬁne compat- ibilit y of sequences of partitions. Let S = ( S 1 , ..., S r ) b e a partition of the set of v ertices of a graph G = ( V G , R G ), and let T = ( T 1 , ..., T s ) b e a partition of the set of v ertices of a graph H = ( V H , R H ). S and T are said to b e c omp atible und er G and H resp ectiv ely if |S | = |T | (i.e. r = s ), and for all i ∈ { 1 , ..., r } , | S i | = | T i | and ADe g ( S i , V G , G ) = ADe g ( T i , V H , H ). Deﬁnition 6 L et G = ( V G , R G ) and H = ( V H , R H ) b e two gr aphs. L et Q G = ( S G , R G , P G ) , and Q H = ( S H , R H , P H ) b e two se quenc es of p artitio ns f or gr aphs G and H r esp e ctively. Q G and Q H ar e said to b e compatible sequences of p artitions if: 1. | S G | = | S H | = t , | R G | = | R H | = | P G | = | P H | = t − 1 . 2. L et R G = ( R 0 G , ..., R t − 1 G ) , R H = ( R 0 H , ..., R t − 1 H ) , P G = ( P 0 G , ..., P t − 1 G ) , P H = ( P 0 H , ..., P t − 1 H ) , S G = ( S 0 , ..., S t ) , S H = ( T 0 , ..., T t ) . F or al l i ∈ { 0 , ..., t − 1 } , R i G = R i H , P i G = P i H , and S i and T i ar e c omp atible under G and H r esp e ctively. 3. L et S t = ( S t 1 , ..., S t r ) , T t = ( T t 1 , ..., T t r ) , then for al l x, y ∈ { 1 , ..., r } , ADe g ( S t x , S t y , G ) = ADe g ( T t x , T t y , H ) . The follo wing theorem sho ws that having compatible sequ ences of partitions is equiv alen t to b eing isomorphic. Theorem 1 ([9]) Two gr aph s G and H ar e isomorphic if and only if ther e ar e two c omp atible se quenc es of p artitions Q G and Q H for gr aphs G and H r esp e ctively. In order to pr op erly handle automorph isms, sequences of partitions will b e extended with v ertex equiv alence information. T w o vertice s u, v ∈ V of a graph G = ( V , R ) are e quivalent , denoted u ≡ v , if there is an automorphism f of G such that f ( u ) = v . A v ertex w ∈ V is ﬁxe d by f if f ( w ) = w . When tw o vertic es are equiv alent, they are said to b elong to the same orbit . The set of all the orbits of a graph is called the orbit p artitio n . Ou r algorithm p erforms a partial compu tation of the orbit p artition. The orbit partition w ill b e computed increment ally , starting from th e s ingleton partition. Since our algorithm p erforms a limited search for automorphisms, it is p ossible that it stops b efore the orbit partition is r eally foun d. Th erefore, w e will in tro duce the notion of semiorbit p artitio n , an d extend the sequence of partitions to includ e a semiorbit partition. Deﬁnition 7 L et G = ( V , R ) b e a gr aph. A semiorbit partition of G is any p artition O = { O 1 , ..., O k } of V , such that ∀ i ∈ { 1 , ..., k } , v , u ∈ O i implies that v ≡ u . Deﬁnition 8 An extended sequence of partitions E for a gr aph G = ( V , R ) is a tuple ( Q , O ) , wher e Q is a se quenc e of p artitions, denote d as Se qPart ( E ) , and O is a semiorbit p art i tion of G , denote d as Orbits ( E ) . Finally , we introd uce a notation for th e num b er of v ertex reﬁn emen ts tagged BA CKT RA CK, since it w ill b e used to c ho ose the target sequence of partitions to b e repro duced. Let Q = ( S , R , P ) b e a sequ en ce of partitions, and let R = ( R 0 , ..., R t − 1 ). Then, Backtr ackAmount ( Q ) = |{ i : i ∈ { 1 , ..., t − 1 } ∧ R i = BA CK T RA CK }| . 5 2.3 Comp onen ts Theorem It was obs erv ed [9 ] that conauto is v ery eﬃcient ﬁnding isomorphism s for unions of str on gly regular graphs, but it is ineﬃcient detecting that tw o suc h unions are not isomorphic. Exploring the b ehavio r of conauto in graphs that are th e d isjoin t union of connected comp onen ts, we observed that it w as not able to identify cases in which comp onents in b oth graph s had already b een matc h ed. This wa s leading to many redun dan t attempts of matc h ing comp onen ts. Note that, once a comp onent C G of a graph G has b een foun d isomorphic to a comp onent C H of a graph H , it is of no use tryin g to matc h C G to another comp onen t of H . Besides, if C G can not b e matc hed to any comp onen t of H , it is of no use trying to matc h the other comp onen ts, since, at the end, the graph s can n ot b e isomorphic. After a thorough stu dy of the b eh a vior of conauto for these graphs, we ha ve concluded that its p erf ormance can b e d rastically impr ov ed in these cases b y d irectly applying the follo wing theorem (whose pr o of can b e found in the App endix): Theorem 2 During the se ar ch for a se quenc e of p artitions c omp atible with the tar get, b acktr acki ng fr om a level l to a level k < l , such that e ach c el l of level l is c ontaine d in a diﬀer ent c el l of level k , c an not pr ovide a c omp atible p artition. 3 Conauto-1.2 In this section we p rop ose a new algorithm conauto-1.2 (d escrib ed in Algorithm 1) which is based on algorithm conauto [9], and u ses the result of Theorem 2 to drastically reduce bac ktrac k in g. It s tarts generating a sequence of partitions for eac h of the graphs b eing tested (using fu nction Gener ateSe quenc eOf P artitions ), and p erforming a limited searc h f or automorphisms using f u nction FindA utomorph i sms , just like conauto. The diﬀerence with conauto is that, during the searc h for the compatible sequence of p artitions ( Match ), the algorithm not alwa ys b ac ktrac ks to the previous recursiv e call (the previous lev el in th e sequence of partitions). In stead, it may bac ktrac k directly to a muc h higher lev el, or ev en stop the searc h, concluding that the graph s are not isomorp h ic, skipping inte r mediate bac k tr ac king p oint s . F unction Gener ateSe quenc eOfPartitions is th e same u sed by conauto (see [9] for th e d etails). It is w orth men tioning that it generates a sequence of p artitions with the follo win g criteria: 1. It starts w ith the degree partition, and ends wh en it gets a p artition in whic h no non-sin gleton cell has remaining links . 2. The pivot c el l used for a reﬁnemen t must alwa ys ha ve r emaining links (the more, the b etter). 3. A t eac h lev el, a v ertex reﬁnement with a singleton p iv ot cell is the preferred c hoice. 4. The second b est c h oice is to p erf orm a set r eﬁnemen t, pr eferring small cells o ver big ones. 5. If the pr evious reﬁnement s can n ot b e us ed , then a v ertex is c hosen fr om the p iv ot cell (the smallest cell with links), a v er tex reﬁnement is p erformed with that pivot vertex , and a bac ktrac kin g p oint arises. F unction FindA utomorphisms is also the same used by conauto (see [9] for the d etails). It tak es as inpu t a sequence of partitions for a graph , and generates an extended sequence of partitions. In the p r o cess, it tries to eliminate bac ktrac kin g p oint s, and builds a semiorbit partition of the ve r tices with the information on vertex equiv alences it gathers. Recall th at tw o vertice s are equiv alen t if there is an automorphism that p ermutes them, i.e., if there are tw o equiv alen t s equences of p artitions in whic h one tak es the place of the other. 6 Algorithm 1 T est whether G and H are isomorphic ( c onauto-1.2 ). Ar eIsomorphic ( G, H ) : b o olean 1 Q G ← Gener ateSe quenc eOfPartitions ( G ) 2 Q H ← Gener ateSe quenc eOfPartitions ( H ) 3 E G ← Fi ndA utomorphisms ( G, Q G ) 4 E H ← Fi ndA utomorphisms ( H , Q H ) 5 if Backtr ackA mount ( Se qPart ( E G )) ≤ Backtr ackA mount ( Se qPart ( E H )) then 6 return 0 ≤ Match (0 , G, H , Se qPart ( E G ) , Orbits ( E H )) 7 else 8 return 0 ≤ Match (0 , H , G, Se qPart ( E H ) , Orbits ( E G )) 9 end if Algorithm 2 Find a sequence of partitions compatible with the target. Match ( l , G, H , Q G , O H ) : integ er 1 if partition labeled FIN and the adjacencies in both partitions match 2 return l 3 else if partition lab eled VER TEX and v ertex reﬁnement compatible then 4 l ′ ← − Match ( l + 1 , G, H , Q G , O H ) 5 if l 6 = l ′ then return l ′ 6 else if partition lab eled SET and set reﬁn ement compatible then 7 l ′ ← − Match ( l + 1 , G, H , Q G , O H ) 8 if l 6 = l ′ then return l ′ 9 else if partition lab eled BA CKTRACK then 10 for each vertex v in the p ivo t cell, while NOT success do 11 if v may NOT b e discarded according to O H and vertex reﬁ n ement compatible then 12 l ′ ← − Match ( l + 1 , G, H , Q G , O H ) 13 if l 6 = l ′ then return l ′ 14 end i f 15 end for 16 end if 17 return the nearest level l ′ such that the condition of Theorem 2 holds F unction Match (Algorithm 2 ) uses bac ktracking attempting to ﬁnd a sequence of partitions for graph H that is compatible with the one for graph G . A t b ac ktrac king p oints, it tries ev ery feasible v ertex in the piv ot cell, so that no p ossible solution is missed. Note that, u nlik e in conauto, the function M atch of conauto-1.2 d o es not return a b o olean, bu t an inte ger. Thus, if Match returns − 1, th at means that a mismatc h has b een f ou n d at some lev el l , suc h th at there is n o pr evious lev el l ′ at wh ic h a cell con tains (at least) tw o cells of the p artition of lev el l . Hence, from Theorem 2 th ere is no other feasible alternativ e in the searc h sp ace that can yield an isomorp hism of the graphs. If it return s a v alue that is higher than the current leve l, then a matc h has b een found, the graph s are isomorphic an d there is no need to con tinue th e searc h . Therefore, in this case the call imm ediately returns with this v alue. If it returns a v alue that is low er than the current level , then it is necessary to b ac ktrac k to that lev el, since trying another option at this lev el is meaningless according to Theorem 2. Hence the algorithm also return s immediately with that v alue. I f a call at level l return s l , then another alternativ e at this lev el l should b e tried if p ossible. In an y other case, it applies Theorem 2 directly , and return s the closest (previous) lev el l ′ at whic h tw o cells of the current lev el l b elong to the same cell of l ′ . If no su c h p revious level exists, it returns − 1. 7 4 P erformance Ev aluation In this section w e compare the p ractical p erformance of conauto-1.2 with naut y and bliss, tw o w ell- kno wn algorithms that are considered the fastest algorithms for isomorphism testing and canonical lab eling. In the p erformance ev aluation exp eriment s , w e hav e r un these programs with instances (pairs of graphs) that b elong to sp eciﬁc families. W e also use conauto to sh o w the improv emen t ac hiev ed b y conauto-1.2 for these graph families. Undirected an d directed (wh en p ossible) graphs of d iﬀeren t sizes (num b er of n o des) h a ve b een considered. The exp erimen ts include ins tances of isomorphic and non-isomorphic pairs of grap h s. 4.1 Graph F amilies F or the ev aluation, we h a ve b uilt some f amilies of graphs with regularly-connecte d comp onent s . The general construction tec hniqu e of these graphs consists of combining small comp onents of diﬀeren t typ es by either (1) connecting every vertex of eac h compon ent to all the v ertices of the other comp on ents, (2) connecting only some vertice s in eac h comp onen t to some v ertices in all the other comp onents, or (3) applying the latter construction in t w o lev els. The us e of these tec hn iques guaran tees th at the resulting graph is connected, whic h is con venien t to ev aluate algorithms that require connectivit y (lik e, e.g ., vf2 [4]). Usin g th e d isjoin t u n ion of connected comp on ents yields similar exp erimental resu lts. Next, we d escrib e eac h family of graphs used. In fact, as the reader w ill easily infer, th e key p oint in all these constructions is that the comp onen ts are either disconnected, or connected via complete n -partite graphs . Hence, multiple other constructions may b e used w hic h w ould yield similar r esu lts. In eac h graph family , one h u ndred p airs of isomorp hic and non-isomorphic graphs ha ve b een generated for eac h graph size (up to appr oximate ly 1 , 000 v ertices). Unions of Strongly Regular Graphs This graph family is built from a set of 20 strongly regular graphs with p arameters (29 , 14 , 6 , 7) as comp onen ts. T he comp onents are interco nnected so that eac h v ertex in one comp onent is connected to every v ertex in the other comp onents. This is equiv alen t to inv erting the comp onen ts, then applying th e disjoint union, and ﬁnally inv erting the resu lt. Grap h s up to 20 × 29 = 580 vertice s h a ve only one cop y of eac h comp onent , and bigger ones ma y h a v e more than one copy of eac h comp onent. Unions of T ripartite Graphs F or this family , w e use th e d igraphs in Figure 1 as the basic comp onent s . F or th e p ositiv e tests (isomorphic graph s) we use the same num b er of comp onen ts of eac h typ e, while for th e negativ e tests we u se one graph with the same n u m b er of comp onents of eac h t yp e, and another graph in whic h one comp onent has b een replaced b y one of the other type. The connectio n s b etw een comp onen ts hav e b een done in the follo wing w ay . Th e v ertices in the A subset of eac h co m p onent are conn ected to all the v ertices in the B su bsets of the other comp onent s . See Figure 1 to lo cate these subsets. The arcs are directed fr om the vertice s in the A subsets, to the v ertices in the B subs ets. F rom the previously describ ed graph s , w e h a v e obtained an undirected v ersion by transformin g ev ery (directed) arc in to an (undirected) edge. Hyp o-Hamiltonian Graphs 2-level-co nnected F or this family w e us e t w o n on -isomorp h ic Hyp o-Hamiltonian graphs with 22 v ertices. Both graphs ha ve fou r orb its of sizes: one, three, six, and tw elv e. These basic comp onents are interconnected at t wo lev els. Let us call the vertic es in 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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A B A B Figure 1: T ripartite graphs us ed as comp onen ts. the orbits of size on e, the 1-orbit vertice s, and the v ertices in the orb its of s ize three the 3-orbit v ertices. In the ﬁrst lev el, we connect n basic comp on ents, to form a ﬁrst-level c omp onent , b y connecting all the 3-orbit ve r tices in eac h basic comp onen t to all the 3-orbit vertices of the other basic comp onen ts. In this construction, the 3-orbit vertices, along with th e new edges added to in terconnect the n basic comp onen ts, form a complete n -partite graph. Th en, in th e second lev el, m ﬁr st-lev el comp onents are inte rconnected by add ing edges that connect the 1-orbit v er tices of eac h ﬁ rst-lev el comp onent with all the 1-orbit v ertices of the other ﬁrst-lev el comp onents. Again, the 1-orbit ve r tices, along w ith the edges connecting them, form a complete m -partite graph. Since w e use t wo Hyp o-Hamiltonian graphs as basic comp onent s, to generate negativ e isomorphism cases, a comp onent of one type is replaced w ith one of the other type. 4.2 Ev alua t ion R esults The p erformance of the f our p rograms has b een ev aluated in terms of their execution time with m u ltiple instances of graph s from the previously deﬁn ed f amilies. The execution times ha v e b een measured in a P entium I I I at 1.0 GHz with 256 MB of m ain memory , u nder Linux RedHat 9 . 0. The same compiler (GNU gcc) and th e same optimization ﬂag (-O) ha ve b een u sed to compile all the programs. The time measured is the real execution time (n ot only C PU time) of the p rograms. This time d o es not include th e time to load the graph s fr om disk int o memory . A time limit of 10 , 000 seconds has b een set for eac h execution. When th e execution of a program with graphs of size s reac hes this limit, all the execution data of that program f or graph s of the s ame family with size no smaller than s are discarded . Av erage Execution Time The r esults of the exp eriments are ﬁ rst pr esen ted, in Figure 2, as curv es that r epresen t execution time as a f unction of graph size. In these curves, eac h p oin t is the a ve r age execution time of the corresp onding program on all the instances of the corresp ondin g size. It w as p reviously kn o wn that naut y requires exp onen tial time to pr o cess graphs that are unions of strongly regular graphs [12]. F rom our results, w e conjecture that bliss has the same p roblem. That do es not app ly to conauto-1.2, though. While the original conauto h ad p roblems with non- isomorphic pairs of graphs, conauto-1.2 ov ercomes this problem. With the family of u nions of tripartite graphs, we ha ve run b oth p ositiv e and negativ e exp eri- men ts w ith directed and undirected v ersions of the graphs. In all cases, conauto-1.2 has a v ery lo w execution time. (A gain, the improv emen t of conauto-1.2 ov er conauto is apparent in the case of negativ e tests.) Observ e that there are no signiﬁcant d iﬀerences in the execution times of bliss and 9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . conauto bliss Avera ge Time [Seconds] 0 Non-Isomorphic Non-Isomorphic Isomorphic 200 Isomorphic Non-Isomorphic Isomorphic Non-Isomorphic 0 400 600 800 1000 1e − 05 Size [ V ertices] conauto-v2 conauto nauty-2.2 bliss Unions of Cubic hypohamilt onian graphs 1e − 05 1e − 04 1e − 04 1e − 03 1e − 02 1e − 01 1e+00 1e − 03 1e+01 1e − 02 1e − 01 1e+00 1e+01 1e+02 1e+02 200 1e+03 0 200 400 1e+03 0 200 600 400 600 800 1000 conauto conauto-v2 nauty-2.2 bliss 800 1000 nauty-adj Avera ge Time [Seconds] bliss Unions of Strongly Regular Graphs conauto conauto-v2 Avera ge Time [Seconds] 0 200 400 600 800 100 0 conauto-v2 conauto nauty-adj bliss Unions of unions of directed triparti te graphs 1e − 05 1e − 04 1e − 03 1e − 02 1e − 01 1e+00 1e+01 1e+02 1e+03 0 200 400 600 800 100 0 Size [V ertices] conauto-v2 400 600 800 100 0 conauto conauto-v2 nauty-2.2 bliss 1e − 05 1e − 04 1e − 03 1e − 02 1e − 01 1e+00 1e+01 1e+02 1e+03 0 200 400 600 800 100 0 conauto-v2 conauto nauty-2.2 bliss Avera ge Time [Seconds] 0 200 400 600 800 1000 conauto-v2 conauto nauty-2.2 bliss Unions of undirected tripartite graphs Isomorphic nauty-2.2 Figure 2: Av erage execution time. conauto-1.2 b et ween the directed and the und irected cases. Ho wev er, nauty is slo wer with d irected graphs, ev en using the adjac enci es in v arian t sp eciﬁcally designed for directed graph s. Our last graph f amily , Cubic Hyp ohamiltonian 2-lev el-connected graphs, h as a more complex structure than the other families, ha ving t wo lev els of interco n nection. Ho wev er, the results do not diﬀer signiﬁcan tly from the previous ones. It seems that these graphs are a b it easier to pro cess (compared with th e other graph families) for bliss, but n ot for nauty . Lik e in the previous cases, conauto-1.2 is fast and consisten t with the graph s in this family . It clearly improv es the results of conauto for the non-isomorphic pairs of graphs. 10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200 400 600 800 1000 Size [V ertices] conauto-v2 conauto nauty-2.2 bliss Normalized Standard Deviation Unions of cubic hyp ohamiltonian graphs Isomorphic Non-Isomorphic 0 0.5 1 1.5 2 0 200 400 600 800 1000 conauto-v2 conauto nauty-adj bliss 0 200 400 600 800 1000 conauto-v2 conauto nauty-adj bliss Normalized Standard Deviation Unions of directed tri partite graphs Isomorphic Non-Isomorphic 0 0.5 1 1.5 2 0 200 400 600 800 100 0 Size [V ertices] conauto-v2 conauto nauty-2.2 bliss 0 Figure 3: Normalized Standard Deviation of execution times. Standard Deviation In addition to the a v erage b ehavio r for eac h graph size, we ha v e also ev aluate d the regular b eha vior of the programs. With regular b eha vior w e mean that th e time required to p ro cess any p air of graphs of the same family an d size is v ery similar. W e ha ve observ ed th at conauto-1.2 is not only fast for all these families of graphs, but it also has a very regular b eha vior. Ho wev er, that do es n ot hold for naut y nor bliss. This is illustrated with the p lots of the normalize d standar d deviation 4 (NSD) sh o wn in Figur e 3. Algorithm conauto-1.2 h as a NSD that remains almost constan t, and ve ry close to cero, for all graph sizes, and even decreases for larger graphs. Ho w ev er, nauty and bliss h a ve a m uch more err atic b eha vior. In the case of conauto, w e s ee that its problems arise when it faces negativ e tests, where the NSD rapidly gro ws. 5 Conclusions an d F utur e W ork W e ha ve presen ted a result (the Comp onen ts Theorem, Theorem 2) that can b e applied in GI algorithms to eﬃcien tly ﬁ nd automorph isms. Th en, we ha ve applied this resu lt to transf orm the algorithm conauto into conauto-1.2. Algorithm conauto-1.2 has b een sho wn to b e fast and consis- ten t in p erforman ce for a v ariet y of graph families. Ho wev er, the algorithm conauto-1.2 can still b e impro ved in sev eral wa ys: (1) b y addin g the capabilit y of computing a complete set of generators for the automorphism group, (2) by making extensiv e use of disco vered automorphisms d u ring the matc h pro cess, and (3) by computing canonical forms of graphs. In all these p ossible impr o ve m en ts, the Comp onent s Theorem w ill surely help. Additionally , th e Comp onen ts Theorem migh t also b e used by nau ty and bliss to improv e their p erformance for the graph families considered, at lo w cost. 4 The normalized standard deviation is obtained by dividing th e standard deviation of the sample by the mean. 11 References [1] Magdy S . Abadir and Jac k F erguson. An impro v ed la yout v eriﬁcation algorithm (LA V A). In EURO-D AC ’90: Pr o c e e dings of the c onfer enc e on Eur op e an design automation , pages 391– 395, Los Alamitos, CA, USA, 1990. IEE E Comp uter So ciet y P r ess. [2] Alfred V. Aho, John E. Hop croft, and Jeﬀrey D. Ullman. The Desig n and Analysis of Computer Algor i thms . Addison -W esley series in computer science and information pr o cessing. Addison- W esley Publish in g Company , Boston, MA, USA, 1974. [3] Donatello Conte , Pa s q u ale F oggia, Carlo Sansone, and Mario V ento. Graph matc hing app li- cations in p attern r ecognition and image p ro cessing. 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Linear time algorithm for isomorp hism of planar graphs (preliminary rep ort). In STOC ’74: Pr o c e e dings of the sixth annual ACM symp osium on The ory of c omputing , pages 172–1 84, New Y ork, NY, USA, 1974. A C M Press. [8] T ommi A. Junttila and P etteri Kaski. Engineering an eﬃcient canonical lab eling tool for large and sparse graphs. In ALENEX . SI AM, 2007. [9] Jos ´ e Lu is L´ op ez-Presa and Anto n io F ern´ andez Anta . F ast algorithm for graph isomorph ism testing. In Jan V ahrenhold, editor, SEA , v olum e 5526 of L e ctur e N otes in Computer Scienc e , pages 221–23 2. Sprin ger, 2009. [10] Brendan D. McKa y . Practical graph isomorphism. Congr essus Numer antium , 30:45– 87, 1981. [11] Brendan D. McKa y . The naut y page. C omp uter S cience Departmen t, Aus tr alian National Univ ersity , 2004. http:// cs.an u.edu .au/ ∼ b dm/nauty/. [12] T akunari Miy azaki. T he complexit y of McKa y ’s canonical lab eling algorithm. In Larry Finkel- stein an d Willi am M. Kanto r , editors, Gr oups and Computa tion II , v olume 28 of DIMACS Series in D i scr ete Mathematics and The or etic al Computer Scienc e , pages 239–256. American Mathematica l So ciet y , Pro vidence, Rho de Island , USA, 1997. [13] Jos ´ e Luis L ´ op ez Pr esa. Eﬃcient Algor ithms f or Gr aph Isomorph ism T esting . Do ctoral thesis, Escuela T´ ecnica Sup erior de In genier ´ ıa d e T elecom unicaci´ on, Universidad Rey J u an Carlos, Madrid, Spain, Marc h 2009. Av aila b le at http://www.diatel. upm.es/jllop ez/tesis/thesis.p df . 12 [14] Johannes Singler. Graph isomorphism imp lemen tation in LED A 5.1. T ec h nical rep ort, Algo- rithmic Solutions Softw are Gm b H, Dec. 2005. [15] Gottfried Tinhofer a n d Mikhail Klin. Alg eb r aic com binatorics in mathematic al c h em- istry . Metho ds and algorithms I I I. Graph in v arian ts and s tabilization metho ds. T ec h- nical Rep ort TUM-M9902, T ec hnisc he Univ ers it¨ at M ¨ unc h en, Marc h 1999. h ttp://www- lit.ma.tum.de/v ero eﬀ/quel/990.0 5005.p df. [16] J. R. Ullmann . An algorithm for sub graph isomorph ism. Journal of the ACM , 23(1):31–42 , 1976. [17] T ak ashi W ashio and Hiroshi Moto da. State of the art of graph-based data m in ing. ACM SIGKDD Explor ations N e wsletter , 5(1):5 9–68, 2003. 13 A Pro of of the Comp onen ts Th eorem (Theorem 2) The follo wing deﬁnition will b e needed in the pro of. Deﬁnition 9 L et G = ( V , R ) b e a g r aph. L et V ′ ⊆ V . Then the su bgraph ind uced by V ′ on G , denote d G V ′ , i s the gr aph H = ( V ′ , R ′ ) such that R ′ = { ( u, v ) : u, v ∈ V ′ ∧ ( u, v ) ∈ R } . A bac ktrac kin g p oin t arises when a partition do es not ha ve sin gleton cells (suitable for a v ertex reﬁnement) and it is not p ossible to r eﬁne su c h partition by means of a set reﬁnement . Let us in tro duce a new concept that will b e useful in the follo wing discussion. Deﬁnition 10 L e t G = ( V , R ) b e a gr aph, and let S = ( S 1 , ..., S r ) b e a p artition of V . S is said to b e equitable (with r esp e ct to G ) if for al l i ∈ { 1 , ..., r } , for al l u, v ∈ S i , for al l j ∈ { 1 , ..., r } , ADe g ( u, S j , G ) = ADe g ( v , S j , G ) . Observ a tion 1 The p artition at a b acktr acking p oint is e quitable. Pro of: Ass u me otherwise. Then, there exists some S j suc h that there are t wo ve rtices u, v in some S i , s uc h that A De g ( u, S j , G ) 6 = ADe g ( v, S j , G ). T herefore, it wo u ld b e p ossib le to p erform a set reﬁnement on the partition, us ing S j as the pivot cell, and vertice s u and v would b e distinguish ed b y this r eﬁnemen t, and cell S i w ould b e sp lit. This is n ot p ossible since, at a bac ktrac king p oin t, no set reﬁnement has su cceeded. Observ a tion 2 L et l b e a b acktr acking level. L et S l = ( S l 1 , ..., S l r ) b e the p artition at that level. Then, for al l i ∈ { 1 , ..., r } , G S l i is r e gular. Pro of: F rom Observ ation 1, S l is equitable. Fix i ∈ { 1 , ..., r } , then, from Deﬁnition 10, for all u, v ∈ S l i , ADe g ( u, S l i , G ) = ADe g ( v , S l i , G ). Th erefore, G S l i is regular, for all i ∈ { 1 , ..., r } . Let Q = ( S , R , P ) b e a sequence of p artitions for graph G = ( V , R ) wh ere S = ( S 0 , ..., S t ), R = ( R 0 , ..., R t − 1 ), and P = ( P 0 , ..., P t − 1 ). F or all i ∈ { 0 , ..., t } let S i = ( S i 1 , ..., S i r i ), and V i = S r i j =1 S i j . W e consider t w o bac ktrac kin g lev els k and l that satisfy th e preconditions of Th eorem 2, i.e., k < l and eac h cell of S l is con tained in a d iﬀeren t cell of S k . Let p ∈ S k P k b e th e p iv ot v ertex used for the vertex r eﬁnemen t at leve l k . Assume there is a vertex q ∈ S k P k , q 6 = p that s atisﬁes the follo wing. T k +1 = V ertexR eﬁnement ( S k , q , G V k ) is a partition that is compatible w ith S k +1 . Let T k +1 = ( T k +1 1 , ..., T k +1 r k +1 ), W k +1 = S r k +1 j =1 T k +1 j . F or all i ∈ { k + 2 , ..., l } , let T i = ( T i 1 , ..., T i r i ) b e compatible w ith S i , where W i = S r i j =1 T i j , T i = SetR eﬁnement ( T i − 1 , T i − 1 P i − 1 , G W i − 1 ) if R i − 1 = SET, and T i = V ertexR eﬁnement ( T i − 1 , v , G W i − 1 ) for some v ∈ T i − 1 P i − 1 if R i − 1 6 = SET. This generates an al ternativ e sequence of partitions th at is compatible with the original one up to lev el l . Under these p remises, w e show in the r est of the sectio n that G V l and G W l are isomorphic, and th ere is an isomorph ism of th em that matc hes the ve r tices in S l i to the v er tices in T l i for all i ∈ { 1 , ..., r l } . T o s implify th e notation, let us assu me r k = r l = r . Note that in this case, for all i ∈ { 1 , ..., r } , S l i ⊆ S k i . In case r k 6 = r l this corresp ond ence is not trivial. Ho wev er, we can safely assu me that 14 there ma y b e some S l i ∈ S l that are emp t y , and develo p our argumen t considering this p ossibility , although w e kno w that in the real s equence of partitions, these empt y cells w ould ha ve b een discarded. F or all i ∈ { 1 , ..., r } , let E i = S k i \ S l i , E ′ i = S k i \ T l i b e the v ertices discarded in the reﬁnements from S k i to S l i and T l i resp ectiv ely , let A i = E i ∩ E ′ i b e the vertic es discarded in b oth alternativ e reﬁnements, B i = E i \ A i the vertic es discarded only in th e reﬁn emen t from S k i to S l i , C i = E ′ i \ A i the v ertices discarded only in the r eﬁnemen t f rom S k i to T l i , and D = S l i ∩ T l i the v ertices remaining in b oth alternativ e partitions at leve l l . Let A = S r i =1 A i , B = S r i =1 B i , C = S r i =1 C i , D = S r i =1 D i , E = S r i =1 E i , and E ′ = S r i =1 E ′ i . Clearly , E = A ∪ B , and E ′ = A ∪ C . Observ e that | E i | = | E ′ i | , and hence | B i | = | C i | for all i ∈ { 1 , ..., r } . E ′ 1 T l 1 E 1 A 1 B 1 S l 1 C 1 D 1 . . . E ′ r T l r E r A r B r S l r C r D r Figure 4: P artition of S k i in to subsets A i , B i , C i , and D i for all i ∈ { 1 , ..., r } . Observ a tion 3 G E is isomorphic to G E ′ , and ther e is an isomorphism of them that matches the vertic es in E i to those in E ′ i , f or al l i ∈ { 1 , ..., r } . Pro of: Direct from the constru ction of the sequences of partitions. Lemma 1 L et M = A dj ( G ) . It i s satisﬁe d that: • F or e ach u ∈ E , for al l i ∈ { 1 , ..., r } , for al l v , w ∈ S l i , M uv = M uw and M vu = M w u . • F or e ach u ∈ E ′ , for al l i ∈ { 1 , ..., r } , f or al l v , w ∈ T l i , M uv = M uw and M vu = M w u . Pro of: Since none of the v ertices in E has b een able to d istinguish among the v ertices in cell S l i , eac h of the discarded v er tices has th e same typ e of adjacency with all the vertic es in S l i . O therwise, consider vertex u ∈ E . Assum e u has at least tw o diﬀeren t t yp es of adjacency with the ve rtices in S l i . Sin ce it w as discarded du ring the reﬁnements from S k i to S l i , that h ad to b e for one of the follo wing reasons: 1. It was discarded for ha vin g no lin k s (i.e. links of type 0), what is imp ossib le since it has t wo diﬀeren t t yp es of adjacencies with the ve r tices in S l i . 2. It was us ed as the pivot set in a v ertex reﬁnemen t, w h at is im p ossible sin ce it w ould hav e b een able to split cell S l i . The same argument applies to the vertices in E ′ with resp ect to the vertic es in eac h cell T l i . Consider the adjacency b et wee n vertex u an d v ertex v is M uv = a f or some a ∈ { 0 , ..., 3 } . T hen, w e will denote the adjacency b et ween v and u ( M vu ) as a − 1 . Note that if a = 0, a − 1 = 0, if a = 1, a − 1 = 2, if a = 2, a − 1 = 1, and if a = 3, a − 1 = 3. Lemma 2 F or e ach i, j ∈ { 1 , ..., r } , ther e is some a ∈ { 0 , ..., 3 } suc h that for al l u ∈ B i , v ∈ C i , w ∈ D i , u ′ ∈ B j , v ′ ∈ C j , and w ′ ∈ D j , M uv ′ = M uw ′ = M vu ′ = M vw ′ = M w u ′ = M w v ′ = a and M u ′ v = M u ′ w = M v ′ u = M v ′ w = M w ′ u = M w ′ v = a − 1 . 15 Pro of: Let us take any i ∈ { 1 , ..., r } and an y j ∈ { 1 , ..., r } . S ince B i ⊆ E and C j ⊆ S l j , from Lemma 1, for eac h u ∈ B i , for all v ′ ∈ C j , M uv ′ = a for some a ∈ { 0 , ..., 3 } . Let us take any su c h v ′ ∈ C j . Then, M v ′ u = a − 1 for those particular v ′ and u . Besides, since C j ⊆ E ′ and B i ⊆ T l i , from Lemma 1, for all u ∈ B i , M v ′ u = b for some b ∈ { 0 , ..., 3 } . Since we already know that M v ′ u = a − 1 for that particular pair of v ertices, then we conclude that for all u ∈ B i , v ′ ∈ C j , M uv ′ = a and M v ′ u = a − 1 , for some a ∈ { 0 , ..., 3 } . S l j = C j ∪ D j and B i ⊆ E . Since for all u ∈ B i , v ′ ∈ C j , M uv ′ = a and M v ′ u = a − 1 , then from Lemma 1, for all u ∈ B i , w ′ ∈ D j , M uw ′ = a (clearly , the same a ) and M w ′ u = a − 1 . T l i = B i ∪ D i and C j ⊆ E ′ . Sin ce for all u ∈ B i , v ′ ∈ C j , M uv ′ = a and M v ′ u = a − 1 , then f r om Lemma 1, for all v ′ ∈ C j , w ∈ D i , M v ′ w = a − 1 and M w v ′ = a (clearly , the same a ). F urthermore, all the vertice s in S l j = C j ∪ D j ha ve the same num b er of adjacen t ve r tices of eac h t yp e in E i = A i ∪ B i . Otherwise, they w ould hav e b een distinguished in th e r eﬁnemen t pro cess from S k to S l . L ikewise, all the v ertices in T l j = B j ∪ D j ha ve the same num b er of adjacen t v ertices of eac h type in E ′ i = A i ∪ C i . Otherwise, they would hav e b een distinguished in the reﬁnement pro cess from S k to T l . Hence, the v er tices of D j m u s t ha ve the same n u m b er of adjacen t v ertices of eac h t yp e in B i and C i . Hence, since for all w ′ ∈ D j , and for all u ∈ B i , M uw ′ = a and M w ′ u = a − 1 , then for all w ′ ∈ D j , and for all v ∈ C i , M vw ′ = a and M w ′ v = a − 1 to o. A similar argument may b e u sed to pro ve that for all w ∈ D i , and for all u ′ ∈ B j , M w u ′ = a and M u ′ w = a − 1 . Then, fr om Lemma 1, since B j ⊆ E , for all u ′ ∈ B j , M u ′ x = M u ′ y for all x, y ∈ S l i . W e already kno w that for all u ′ ∈ B j , M u ′ w = a − 1 for all w ∈ D i , and S l i = C i ∪ D i . Hence, for all v ∈ C i , M u ′ v = a − 1 to o, and M vu ′ = a . Putting together all the partial results obtained, we get the assertion s tated in the lemma. Corollary 1 L et M = A dj ( G ) . F or e ach i ∈ { 1 , ..., r } , it is satisﬁe d that for al l u ∈ B i , v ∈ C i , w ∈ D i , M uv = M vu = M uw = M w u = M vw = M w v = a , wher e a ∈ { 0 , 3 } . Pro of: F rom Lemma 2, for the case i = j , w e get that f or all u ∈ B i , v ∈ C i , w ∈ D i , M uv = M uw = M vu = M vw = M w u = M w v = a and M uv = M uw = M vu = M vw = M w u = M w v = a − 1 . Hence, it m u st hold that a = a − 1 , so a ∈ { 0 , 3 } . Let us deﬁn e t wo families of partitions of A i for i, j ∈ { 1 , ..., r } : A cj i = { x ∈ A i : ∀ u ∈ B i , v ′ ∈ C j , M xv ′ = M uv ′ } A nj i = { x ∈ A i : ∀ u ∈ B i , v ′ ∈ C j , M xv ′ 6 = M uv ′ } Note that, since the v ertices of A i are unable to distinguish among the v er tices of C j , then, if M xv ′ 6 = M uv ′ for some u ∈ B i or some v ′ ∈ C j , then M xv ′ 6 = M uv ′ for all u ∈ B i and all v ′ ∈ C j . Hence, eac h pair of sets A cj i and A nj i deﬁnes a p artition of A i . Note also that, since eac h v ertex in A i has the same t yp e of adjacency with all the vertic es in B i ∪ C i ∪ D i (from L emma 1), then for all x ∈ A cj i , u ∈ B i , v ∈ C i , w ∈ D i , u ′ ∈ B j , v ′ ∈ C j , and w ′ ∈ D j , M xu ′ = M xv ′ = M xw ′ = M uv ′ = M uw ′ = M vu ′ = M vw ′ = M w u ′ = M w v ′ (from Lemma 2). Lemma 3 F or al l i ∈ { 1 , ..., r } , let A c i = T r j =1 A cj i , and let A n i = S r j =1 A nj i . Then, any isomor- phism of G E and G E ′ that maps G E i to G E ′ i , maps the v e rtic es i n A n i among themselves. 16 E ′ i T l i E i A n i A c i B i S l i C i D i Figure 5: P artition of A i in to su bsets A c i , and A n i . Pro of: F rom Ob serv ation 1, partition S k is equitable. Hence, for eac h i, j ∈ { 1 , ..., r } , f or all u, v ∈ S k i , ADe g ( u, S k j , G ) = ADe g ( v , S k j , G ). Th us, for all x ∈ A cj i , y ∈ A nj i , u ∈ B i , v ∈ C i , w ∈ D i , ADe g ( x, S k j , G ) = ADe g ( y , S k j , G ) = ADe g ( u, S k j , G ) = ADe g ( v , S k j , G ) = ADe g ( w, S k j , G ). Let us tak e any pair of v alues of i and j . F rom Lemma 2, all the vertic es of B i ha ve the same t yp e of adjacency with all the v ertices of S l j = C j ∪ D j . Assume this type of adjacency is a . F rom the deﬁn ition of A cj i , all the v ertices of A cj i ha ve adjacency a with all the v ertices of S l j . Hence, for x ∈ A cj i , u ∈ B i , ADe g ( x, S l j , G ) = ADe g ( u, S l j , G ). Since ADe g ( x, S k j , G ) = ADe g ( u, S k j , G ) and ADe g ( x, S l j , G ) = AD e g ( u, S l j , G ), then A D e g ( x, E j , G ) = AD e g ( u, E j , G ) (note th at E j = A ci j ∪ A ni j ∪ B j , S l j = C j ∪ D j , and S k j = E j ∪ S l j ). Ho we ver, from the deﬁnition of A nj i , for y ∈ A nj i , ADe g ( y , S l j , G ) 6 = ADe g ( x, S l j , G ). Hence, since ADe g ( y , S k j , G ) = ADe g ( x, S k j , G ), A De g ( y, E j , G ) 6 = ADe g ( x, E j , G ). Since an y isomorp hism must matc h v ertices with the same degree, every isomorphism of G E and G E ′ that m aps G E i to G E ′ i , maps the v er tices in A nj i among themselv es. Applying this argument o ve r all p ossible v alues of j , we get that an y isomorph ism of G E and G E ′ that maps G E i to G E ′ i , maps the v er tices in A n i among themselv es, for all i ∈ { 1 , ..., r } . Let us fo cus on an y isomorphism of G E and G E ′ that maps G E i to G E ′ i for all i ∈ { 1 , ..., r } (there is at least one fr om Observ ation 3). Lemma 4 G B is isomorp hic to G C , and ther e is an isomorp hism of them that matches the vertic es in B i to those in C i , f or al l i ∈ { 1 , ..., r } . Pro of: Let us analyze the adjacencies b et ween the v ertices in A c i , B i , C i , A c j , B j , and C j for some v alues of i and j . F rom C orollary 1, f or all u ∈ B i , v ∈ C i , M uv = M vu = a , wh ere a ∈ { 0 , 3 } . F rom the deﬁnition of A c i , for all x ∈ A c i , M xu = M xv = M ux = M vx = M uv = a . F rom Lemma 3, the v er tices of A n i are mapp ed among themselve s in any isomorph ism of G E and G E ′ that maps G E i to G E ′ i . Hence, the v ertices of A c i ∪ B i m u s t b e mapp ed to th e v ertices of A c i ∪ C i . If a = 0, then A c i , B i , and C i are disconnected. Hence, G B i and G C i m u s t b e isomorp hic. In the case a = 3, taking the in verses of the graphs leads to the same r esu lt. F rom L emma 2, for eac h i, j ∈ { 1 , ..., r } , there is some a ∈ { 0 , ..., 3 } suc h th at for all u ∈ B i , v ∈ C i , u ′ ∈ B j , v ′ ∈ C j , M uv ′ = M vu ′ = a and M u ′ v = M v ′ u = a − 1 . F rom the deﬁnition of A c i , for all x ∈ A c i , for all u ∈ B i , v ∈ C i , u ′ ∈ B j , v ′ ∈ C j , M xu ′ = M xv ′ = M uv ′ . Putting all this together, we come to a p ictur e of th e adjacencies among A c i , B i , C i , A c j , B j , and C j as sho wn in Figure 6. The connections b etw een the v ertices of A c i and the v ertices of B i , and b et w een the v ertices of A c i and th e v ertices of C i are all-to-all (all the same) of v alue 0 or 3. Similarly , the adjacencies b et ween the v ertices of A c j and the ve r tices of B j , and the adj acencies 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A c j A c i B i B j A c j A c i C j C i Figure 6: Adjacencies b et we en E i and E j , and b et ween E ′ i and E ′ j . b et ween the v ertices of A c j and the v ertices of C j are all th e same, all-to -all 0 or 3 (not n ecessarily equal to those of A c i and B i or C i ). The adj acencies b et ween A c i and B j ∪ C j are all the same, all-to- all of an y v alue in th e set { 0 , ..., 3 } . This also applies to the adjacencies b et ween A c j and B i ∪ C i . If G B i ∪ B j is not isomorphic to G C i ∪ C j , the discrepancy m ust b e in the adjacencies b et wee n v ertices of B i and B j with resp ect to the adjacencies b et wee n ve r tices of C i and C j . In suc h a case, in the isomorphism b et wee n G E i ∪ E j and G E ′ i ∪ E ′ j (recall that from Observ ation 3 there is an isomorphism of G E and G E ′ that m aps the vertic es of E i to the vertic es in E ′ i for all i ∈ { 1 , ..., r } ) some v ertices of A c i should b e mapp ed to vertice s of C i , and some of the v ertices of B i should b e mapp ed to v ertices of A c i . Ho wev er, d u e to the adj acencies among A c i , B i , C i , A c j , B j , and C j , sho wn in Figure 6, th at wo u ld imp ly that the adjacencies b etw een the vertic es of B i and B j had to matc h adjacencies b et we en the v ertices of A i c and A j c . But, in that case, the same adjacency pattern m ust exist b et ween the v ertices of C i and C j , to matc h the corresp onding subgraph of G E i ∪ E j . Hence, the adjacencies b et ween B i and B j could hav e b een matc hed to the adjacencies b et ween C i and C j . Since this applies for all v alues of i and j , we conclude that G B is isomorph ic to G C , and there is an isomorphism of them that matc h es the vertic es in B i to those in C i , for all i ∈ { 1 , ..., r } , completing the pro of. Lemma 5 G V l and G W l ar e isomorphic, and ther e is an isomo rphism of them that maps the vertic es in S l i to the v ertic es of T l i for al l i ∈ { 1 , ..., r } . Pro of: F rom Lemma 2, w e kn o w that for eac h i, j ∈ { 1 , ..., r } , there is s ome a ∈ { 0 , ..., 3 } such that f or all u ∈ B i , v ∈ C i , w ∈ D i , u ′ ∈ B j , v ′ ∈ C j , and w ′ ∈ D j , M uv ′ = M uw ′ = M vu ′ = M vw ′ = M w u ′ = M w v ′ = a and M u ′ v = M u ′ w = M v ′ u = M v ′ w = M w ′ u = M w ′ v = a − 1 . Note also that, from Corollary 1, f or all u ∈ B i , v ∈ C i , w ∈ D i , M uv = M vw = M w u = a , where a ∈ { 0 , 3 } . This adjacency pattern is graph ically sho wn in Figure 7. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B j B i D i D j C j C i D j D i Figure 7: Adjacencies b etw een S l i and S l j , and b et ween T l i and T l j . 18 F rom Lemma 4, we kn o w that G B is isomorphic to G C , and there is an isomorphism of them that matc hes th e v ertices in B i to those in C i , for all i ∈ { 1 , ..., r } . F rom the fact that G D is isomorphic to itself, and the previous consid erations on the adjacency pattern b et wee n the vertice s in B i , C i , D i , B j , C j , an d D j for all i, j ∈ { 1 , ..., r } , shown in Fig- ure 7, it is easy to see th at the isomorp hism of G B and G C obtained from Lemma 4, toghether with the trivial automorphism of G D yields an isomorph ism of G V l and G W l , wh at completes the pro of. W e ha ve sho wn that if t w o alternativ e sequences of partitions S k +1 , ..., S l and T k +1 , ..., T l lead to compatible partitions S l and T l , where all their cells are sub cells of diﬀeren t cells of a previous common lev el k , th en the remaining subgraphs are isomorph ic, and the vertices in eac h cell of one partition m a y b e mapp ed to the vertice s in its corresp on d ing cell in the other p artition by one suc h isomorphism. Th u s, if dur ing the search for a sequence of partitions compatible with the target, w e ha ve got an incompatibilit y at some p oint b ey on d lev el l , and we h a v e to b acktrac k from one lev el l to another lev el k in whic h all the ce lls are diﬀerent sup ersets of the cells in the current bac ktrac kin g p oin t, when trying a compatible path, w e w ill get to the same d ead-end. Hence, it is of no us e to tr y another p ath from one su ch leve l k , and it w ill b e necessary to bac ktrac k to some p oint wh ere at least tw o cells in th e current bac ktr acking p oin t are s ubsets of the same cell in the previous bac ktracking p oint . This prov es Theorem 2. 19

Fast Isomorphism Testing of Graphs with Regularly-Connected Components

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