Construction Sequences and Certifying 3-Connectedness

Symposium on Theoretical Aspects of Computer Science 2010 (Nancy , Fr ance), pp. 633-644 www .st acs-conf .org Construction Sequences and Certifying 3 -Connectedness JENS M. SCHMIDT Dept. of Computer Science, F reie Univers it¨ at, Berlin, German y E-mail addr ess : jens.schmi dt@inf.fu-berlin.de Abstra ct. T utte prove d that every 3-connected g raph on more than 4 n o des has a c on- tr actible e dge . Barnette and Gr ¨ unbaum prov ed the existence of a r emovable e dge in the same setting. W e show that the sequ e nce of contracti ons and the sequence of remo v als from G to the K 4 can b e computed in O ( | V | 2 ) time by extending Barnette and Gr ¨ unbaum’s theorem. A s an application, w e deriv e a certificate fo r the 3-connectedness of graphs that can b e easily computed and verified. 1. In tro duction Instead of dea ling with contrac tions or remo v als in a 3-connected graph G = ( V , E ) w e tak e the equiv alen t view of starting with the complete graph on four vertices K 4 and applying their inv erse op erations unti l G is constructed. Suc h a sequence is called a c on- struction se quenc e of G . W e will define con tr actions, remo v als and their in verse op erations in Section 2. Although existence th eorems on contrac tible and remov able edges are u sed frequently in graph theory [14, 10, 11], w e are not a ware of an y computational r esults to find the whole construction sequence, except w hen contract ions and r emo v als are allo wed to in termix [1]. Moreo ve r, efficient algorithms are un lik ely to b e d eriv ed from the existence pro ofs as they , e. g., in the case of Barn ette and Gr ¨ unbaum, dep end hea vily on adding longest p aths, w hic h are NP-hard to fin d. In cont rast, w e sho w that it is p ossible to find a construction sequence for a graph G in time O ( | V | 2 ) for Barnette and Gr ¨ un baum’s c haracterization, at the exp ense of ha ving parallel edges in in termediate graphs . In addition, we sh o w that Barnette and Gr ¨ un baum’s sequ ence can b e trans f ormed in linear time to T utte’s sequ ence of con tractions and is therefore algorithmically at least as p o werful. Both algorithms do not rely on the 3-connectedness test of Hop croft and T arjan [6], w h ic h runs in linear time b ut is r ather in v olv ed. Blum and Kannan [3] introdu ced the concept of c ertifying algorithms , whic h give an easy-to-v erify p r o of of correctness al ong with their output. While being imp ortan t for program verificati on, ce rtifying algorithms pro vide often new insigh ts in to a p roblem, whic h Key wor ds and phr ases: A lgorithms and data structures, construction sequence, 3-connected, certifying algorithm, T utte contraction, remov able edges, ACM classific ation : F.2.2;G.2.2. This researc h w as supp orted by the Deutsche F orsc hungsge meinschaf t within th e researc h t raining group “Methods for Discrete S tructures” (GRK 1408). c  J. M. Schmidt CC  Creative Commons Attribution- NoDerivs License 634 J. M. SCHMIDT can lead to n ew metho ds . F or that r easons they are a ma jor goal for p roblems on which the fast solutions kn o w n are complicated and d ifficult to implement. T esting a graph on 3-connectedness is such a problem, bu t sur p risingly few w ork has b een devot ed to certifying algorithms, although a sophisticated linear-time algorithm w ithout certificates is kn o wn for o v er 35 yea rs [6, 15, 16]. In fact, w e are aw are of only one certifying algorithm f or that problem [1], w hic h runs in quadratic time, b ut is q u ite in v olv ed. Using construction sequences, we giv e a simple, alternativ e solution with runn in g time O ( | V | 2 ) and sh o w that the used certificate is easy to verify in time O ( | E | ). W e first recapitulate we ll-kno wn results on the existence of constr u ction sequences in Sections 2.1 and 2.2 and p oint out ho w T utte’s sequence can b e obtained from Barnette and Gr ¨ unbaum’s s equ ence in linear time. Sections 2.3 and 3 co v er the main idea for th e existence result that w e u se for computing Barnette and Gr ¨ unbaum’s sequence. Secti on 4 deals w ith the question ho w construction sequences are efficient ly repr esen ted and Section 5 sho ws ho w to use construction sequ ences for a certifying 3-connectedness test. 2. Construction Sequences Let G = ( V , E ) b e a finite graph with n := | V | , m := | E | , V ( G ) = V and E ( G ) = E . A graph is c onne cte d if there is a path b et ween an y t w o no des and disc onne cte d otherwise. F or k ≥ 1, a graph is k -c onne cte d if n > k and deleting ev ery k − 1 no d es lea v es a connected graph. A no de (a pair of no d es) that lea ves a d isconnected graph up on d eletion is called a c ut vertex (a sep ar ation p air ). Note that k -connectedness do es not dep end on parallel edges nor on self-loops. A path leading from no d e v to no de w is denoted by v → w . F or a no de v in a graph, let N ( v ) = { w | v w ∈ E } d enote its set of neighbors and deg ( v ) its degree. F or a graph G , let δ ( G ) b e the minimum degree of its vertic es. A sub division of a graph r ep laces eac h edge by a path of length at least one. C on v ers ely , w e wan t a notation to get bac k to the graph without su b d ivid ed edges. If deg ( v ) = 2, | N ( v ) | = 2 and v / ∈ N ( v ) for a graph G , let smo oth v ( G ) b e the graph obtained from G by deleting v follo wed by ad d ing an edge b et w een its neigh b ors; w e sa y v is smo othe d . I f one of the conditions is violated, let smo oth v ( G ) = G . Let smo oth ( G ) b e the graph obtained b y smo othing eve ry n o de in G . F or an edge e ∈ E , let G \ e denote the graph obtained from G by deleti ng e . Let K n b e the complete graph on n n o des. The follo w ing are we ll-kno w n corollaries of Menger’s theorem [8]. Lemma 2.1. (F an L emma) L et v b e a no de in a gr aph G that is k -c onne cte d with k ≥ 1 and let A b e a set of at le ast k no des in G with v / ∈ A . Then ther e ar e k internal ly no de-disjoint p aths P 1 , . . . , P k fr om v to distinct no des a 1 , . . . , a k ∈ A such that for e ach of these p aths V ( P i ) ∩ A = a i . Lemma 2.2. (Exp ansion Lemma [17 ]) L et G b e a k - c onne cte d gr aph. Then th e gr aph obtaine d by adding a new no de v joine d to at le ast k no des in G is stil l k -c onne cte d. 2.1. T utte’s Characterization a nd their In v erse F rom no w on we assume f or simplicit y that our input graph G = ( V , E ) is simple al- though all results can b e extended to multi graphs. Generally , con tractions cannot alwa ys a void parallel edges in inte rmediate graphs , e. g., for wheels. That is why we define con- tractions to pr eserv e graphs to b e simp le: Contr acting an edge e = xy in a graph deletes e , Construction Sequences and Certifying 3-Connectedn ess 635 iden tifies no des x and y and replaces iterativ ely all 2-c ycles b y an edge. An edge e is called c ontr actible if contrac ting e results in a 3-connected graph . A no de splitting tak es a no de v of a 3-connected graph , replaces v by t wo no des x and y with an edge b et ween them and replaces every former edge uv that wa s inciden t to v with eit her the edge ux , uy or b oth suc h th at | N ( x ) | ≥ 3 and | N ( y ) | ≥ 3 in the n ew graph. No de splitting as d efined here is therefore the exact in v er s e of con tracting a con tractible edge that has on b oth endn o des at least 3 neigh b ors. Theorem 2.3. (Corollary of T utte [13]) The fol lowing statements ar e e quivalent: A simple gr aph G is 3 -c onne cte d ⇔ ∃ se qu enc e of c ontr actions fr om G to K 4 on c ontr actible e dges e = xy with | N ( x ) | ≥ 3 and | N ( y ) | ≥ 3 (2.1) ⇔ ∃ c onstruction se quenc e fr om K 4 to G using no de splittings (2.2) W e describ e next a straigh t-forw ard O ( n 2 ) algorithm to co mpute (2.1) for a graph G on more than 4 v ertices. First, w e decrease the num b er of edges to O ( n ) in G by applying the algorithm of Nagamoc hi and Ib araki [9]. This preserves the 3-connecte dness or resp ectiv ely , the non 3-connecte dness of G . M oreo ver, it is kno w n that the resulting graph con tains a v ertex v of degree 3. By a result of Halin [5], eve ry no de of d egree 3 is inciden t to a contrac tible edge e . W e get e b y subsequently contrac ting eac h of the three inciden t edges and testing the r esu lting graph with the algorithm of Hop croft and T arjan [6] for 3-connectedness. Iteration of b oth su broutines give s us the whole cont raction sequence in O ( n 2 ) time. Ho wev er, the Hop croft-T arjan test is difficult to implement and we will giv e a muc h simpler algorithm that is capable of computin g b oth c haracterizatio ns later. 2.2. Barnet t e and Gr ¨ un baum’s C haracterization and their Inv erse The Barnette and Gr ¨ unbaum op erations ( BG-op er ations ) consist of the follo wing op- erations on a 3-connected graph (see Figures 1(a)-1(c)). (a) add an edge xy (p ossibly a parallel edge) (b) sub d ivide an edge ab by a no de x and add the edge xy for a no de y / ∈ { a, b } (c) sub d ivide tw o distinct, non-parallel edges by no des x and y , resp ectiv ely , and add the edge xy In all three cases, let xy b e the edge that was adde d b y the BG-op eration. (a) parallel edges allow ed (b) y , a , b distinct (c) e 6 = f , e and f not parallel Figure 1: T he three op erations of Barnette and Gr ¨ u n baum. Theorem 2.4. (Barnette and Gr ¨ u n baum [2], T utte [14]) A gr aph G is 3 -c onne cte d if and only if G c an b e c onstructe d fr om the K 4 using BG-op er ations. 636 J. M. SCHMIDT Theorem 2.4 wa s pr o ven in this notation b y Barn ette and Gr ¨ unbaum [2], b ut implicitly describ ed in a th eorem ab out no dal c onne ctivity by T utte [14, Theorem 12 . 65 ]. If not s tated otherwise, ev ery construction sequence uses only BG-op erations. Let a BG-op eration b e b asic , if it do es not create parallel edges and let a constru ction sequence b e b asic , if it only uses basic BG-op erations. Lik e in Th eorem 2.3, w e wa n t the inv erse of a BG-op eration. Let r e moving the edge e = xy of a graph b e the op eration of deleting e f ollo we d by smo othing x and y . An edge e = xy in G is called r emovable , if remo ving e yields a 3-connected graph. W e show that remo v in g a remo v able ed ge e = xy with | N ( x ) | ≥ 3, | N ( y ) | ≥ 3 and | N ( x ) ∪ N ( y ) | ≥ 5 is exactly the inv erse of a BG-op eration. Theorem 2.5. The fol lowing statements ar e e quivalent: A simple gr aph G is 3 -c onne cte d (2.3) ⇔ ∃ se quenc e of r emovals fr om G to K 4 on r emovable e dges e = xy with | N ( x ) | ≥ 3 , | N ( y ) | ≥ 3 and | N ( x ) ∪ N ( y ) | ≥ 5 (2.4) ⇔ ∃ c onstruction se q u enc e fr om K 4 to G using BG-op er ations (2.5) ⇔ ∃ b asic c onstruction se quenc e fr om K 4 to G using BG-op er ations (2.6) Pr o of. Theorem 2.4 establishes (2.3) ⇔ (2.5). Moreo ver, the pro of of T heorem 2.4 in [2] implicitly shows that on simple graph s basic op er ations s u ffice, thus only the equiv alence for (2.4) remains. W e first pr o ve (2.6) ⇒ (2.4) and then (2.4) ⇒ (2.5). BG-op erations operate b y d efinition on 3-connecte d graphs, this h olds in particular for the ones in (2.5). Let G ′ b e the graph obtained by a b asic BG-o p eration in (2.5) that adds the edge e = xy . The op eration can clearly b e undone by remo ving e in G ′ . Since BG-op erations preserve 3-connectedness with Theorem 2.4, | N ( x ) | ≥ 3 and | N ( y ) | ≥ 3 hold in G ′ . It remains to sho w that | N ( x ) ∪ N ( y ) | ≥ 5 in G ′ . If | N ( x ) | ≥ 4 or | N ( y ) | ≥ 4, | N ( x ) ∪ N ( y ) | ≥ 5 follo ws, sin ce x and y are neighbors and no self-loops exist. Thus, let | N ( x ) | = | N ( y ) | = 3. Ha ving N ( x ) \ { y } 6 = N ( y ) \ { x } yields | N ( x ) ∪ N ( y ) | ≥ 5 as well, so let N ( x ) \ { y } an d N ( y ) \ { x } con tain the same t w o no d es a and b . If | V ( G ) | > 4, a or b must b e adjacen t to a n o de c that is neither adjacen t to x nor y . Bu t then { a, b } is a separation p air, con tradicting the 3-connectedness of G . On th e other hand, | V ( G ) | = 4 is not p ossible, since that implies the BG-op eration to b e (a) (since only (b) and (c) create new v ertices) and that is no basic op eration on the K 4 . W e pr o ve (2.4) ⇒ (2.5). Let G ′ b e the graph con taining a remov able edge e = xy that is remo ved in (2.4). Note that G ′ can h a ve p arallel edges due to pr evious remov als but n o self- lo ops. T he remo v al can b e un done by one of the BG-op erations. Whic h one, is dep end en t on the n u m b er i of endno des of e on which smo othing c h anged the graph, i. e., the n umb er of endno des u of e with | N ( u ) | = deg ( u ) = 3 in G ′ . If i = 0, r emo vin g e just deletes e w hic h is inv ersed by op eration (a). F or i = 1, let x b e the n o de with | N ( x ) | = deg ( x ) = 3 in G ′ and f b e the ed ge in w h ic h x w as smo othed. Then (b) can b e app lied, b ecause y / ∈ f (see Figure 8(a)) since otherwise x w ould h a ve had only 2 neigh b ors in G ′ , con tradicting the assumption | N ( x ) | ≥ 3. If i = 2, let f 1 and f 2 b e the edges in which x and y were smo othed. Op eration (c) can only b e applied if f 1 and f 2 are neither iden tical (see Figure 8(b)) nor parallel. But f 1 = f 2 w ould again con tradict | N ( x ) | ≥ 3 in G ′ and f 1 b eing parallel to f 2 w ould con tradicts Construction Sequences and Certifying 3-Connectedn ess 637 | N ( x ) ∪ N ( y ) | ≥ 5 in G , sin ce in that case x and y are only adjacen t to eac h other and the t wo no d es f 1 ∩ f 2 . W e sho w that Barnette and Gr ¨ unbaum’s c haracterization is algorithmically at least as p o w erful as T utte’s by giving a simple linear time transformation. Lemma 2.6 allo w s us to fo cus on computin g BG-op erations only . Lemma 2.6. Every c onstruction se qu enc e usi ng BG-op er ations c an b e tr ansforme d in line ar time to T utte’s se quenc e (2.1) of c ontr actions. Pr o of. W e transform ev ery BG-op eration in r ev er s e order of the construction sequence to 0, 1 or 2 contrac tions eac h. Op eration (a) yields n o contrac tion while op eration (b) yields the con tr action of exactly one part of the su b divid ed edge (either xa or xb in Figure 1). F or an op eration (c), let e = ab and f = v w b e the edges that are sub divided with x and y . Both edges s hare at most one no de; let w. l. o. g. a = v b e that no de if it exists. W e create one con traction f or eac h of the edges xb and y w in arb itrary order. In all cases, con tractions inv erse BG-op erations except for the added edge xy , which is left o v er. But additional edges do not harm the 3-connectedness of the graph nor su bsequent cont ractions. Th us, w e hav e found a con traction sequence to the K 4 unless the fi rst con traction in the case of an op er ation (c) yields at some p oint a graph H that is n ot 3-connected. But H can b e obtained from the graph that results fr om con tr acting th e second edge by app lying one op eration (b) and therefore is 3-connected. 2.3. I den tifying In termediate Graphs with Sub divisions in G Let K 4 = G 0 , G 1 , . . . , G z = G b e the 3-connected graphs obtained in a construction sequence Q to a simple 3-connected graph G using the basic BG-op erations C 0 , . . . , C z − 1 . W e can r ev er s e Q by starting with G and remo ving the add ed edges of BG-op erations in rev erse order. Su p p ose w e wo uld delete th e add ed edge of ev ery C i instead of remo ving it and treat emerging paths con taining inte rior no d es of degree 2 as (top ological) edges in G i (see Figure 2). T hen iterativ ely paths are deleted instead of edges b eing remo v ed and we obtain the sequence of sub divisions G = S z , . . . , S 0 in G with S 0 b eing a sub division of the K 4 . This leads to the follo w ing observ ation. Lemma 2.7 (Observ ation) . L et Q b e a c onstruction se que nc e fr om a gr aph G 0 to G u sing BG-op e r ations. Then G c ontains a sub division of G 0 that is sp e cifie d by Q . In particular, Observ ation 2.7 yields with Theorem 2.4 that ev ery 3-connected graph con tains a sub division of the K 4 (Theorem of J . Isb ell [2]). Eac h graph G i in ou r constru ction sequence can b e identified with the unique sub division S i con tained in G . Con v ersely , G i = smo oth ( S i ) for all 0 ≤ i ≤ z , since smo othing a graph is exactly the in v erse op eration of sub dividin g a graph withou t no des of degree tw o. The no d es x in S i with deg ( x ) ≥ 3 are called r e al no d es, b ecause they corresp ond to no d es in G i . Real no des hav e at least 3 neigh b ors in G i , b ecause G i is 3-connected. Note that in n on -b asic construction sequ en ces smo oth ( S i ) can h a ve parallel edges, al- though S i is alw a ys simple. W e define the links of eac h S i to b e the uniqu e paths in S i with only their endn o des b eing real. The links of S i partition E ( S i ) b ecause S i is 2-connected, has therefore minimum degree tw o and is not a cycle. Let tw o links b e p ar al lel if they share the same endn o des. 638 J. M. SCHMIDT (a) K 4 = G 0 = smo oth ( S 0 ) (b) G 1 = smo oth ( S 1 ) (c) G 2 = smo oth ( S 2 ) (d) G 3 = G (e) S 0 (f ) S 1 (g) S 2 (h) S 3 = G Figure 2: The graphs G 0 , . . . , G z and S 0 , . . . , S z of a construction sequ ence of G . On graph s S i , the d ashed edges and no d es are in G b ut n ot in S i and no des depicted in blac k are r e al no des. F or example, th e p ath C 0 = e → h → g is a BG-p ath for S 0 , yielding S 1 . The links of S 1 are the paths C 0 , a → b → c and the single edges ae , ef , f c , cd , da , f g , g d . Definition 2.8. A BG-p ath for S i is a path P = x → y in G with the follo wing prop erties: (1) S i ∩ P = { x, y } (2) x and y are not b oth contai ned in a link of S i except as end n o des (3) x and y are not inner n o des of links of S i that are p arallel It is easy to see that every BG-path for S i corresp onds to a BG-op eration on G i and vice v ersa. W e w ill exploit this dualit y in th e next section. In general, construction sequences are not b ound to start with the K 4 . Tito v and Kelmans [12, 7] extended Theorem 2.4 by proving the existe nce of a construction sequence ev en wh en starting with arbitrary 3-connected graphs G 0 instead of the K 4 , as long as a sub d ivision of G 0 is con tained in G . This is a generaliza tion, since ev ery 3-co nnected graph con tains a su b division of the K 4 b y Observ ation 2.7. Theorem 2.9. [7, 12] L et G 0 b e a 3 -c onne cte d gr aph. Then a simple gr aph G is 3 -c onne cte d and c ontains a sub division of G 0 if and only if G c an b e c onstructe d fr om G 0 using b asic BG-op e r ations. 3. Prescribing Sub divisions Both Theorems 2.4 and 2.9 c ho ose a v ery sp ecial sub division of the K 4 (resp. G 0 ) on whic h the construction sequence starts, in fact one ha ving the maximum n umber of edges in G . Th e construction sequence is then obtained by adding longest BG-paths. Un fortunately , Construction Sequences and Certifying 3-Connectedn ess 639 computing these dep ends hea vily on solving the longest paths problem, which is known to b e NP-hard ev en for 3-connected grap h s [4]. This giv es rise to the question whether Theorems 2.4 and 2.9 can b e strengthened to start at a pr e scrib e d su b division H ⊆ G of G 0 instead of an arbitrary one. Note that this is equiv alen t to the constraint S 0 = H . Su c h a r esult w ould p ro vide an efficien t computational approac h to construction s equences, since it allo ws us to searc h the n eigh b orho o d of H for BG-paths, yielding a new prescrib ed sub division of a 3-connected graph. Figure 3: Ev ery p ossible BG- op eration adds a parallel edge. Ho wev er, when restricted to basic op erations it is not p ossible to prescrib e H , as the minimal counte rexample in Figure 3 sho ws: Consider the graph G consisting of a K 4 = H d epicted in blac k with an additional no de con- nected to thr ee no des of the K 4 . Then ev ery BG-path for H will create a p arallel link, although G is simple. But what if w e dr op the cond ition that construction sequences ha v e to b e basic? The follo wing theorem sho ws that at this exp ense we can indeed start a construction sequence from an y prescrib ed sub division. Theorem 3.1. L et G b e a 3 -c onne cte d gr aph and H ⊂ G with H b ei ng a sub division of a 3 - c onne cte d gr aph. Then ther e is a BG-p ath for H in G . Mor e over, every link of H of length at le ast 2 c ontains an inner no de on which a BG-p ath for H starts. Pr o of. W e distinguish tw o cases. • H 6 = smo oth ( H ) . Then link s of length at lea st 2 exist in H and we pick an arb itrary one of them, sa y T . Let x b e an inner no de of T , and let Q b e the set of p aths in G from x to a no de in V ( H ) \ V ( T ) av oiding the endno des of T (see Figure 5). By the 3-c onnectedness of G , the set Q cannot b e empt y and ev ery path in Q fulfills Definition 2.8.2. T here is at least one path P = x → y in Q with y b eing not contai ned in a parallel link of T , b ecause otherwise the endn o des of T w ould form a separation pair. Let x ′ b e the last no d e in P that is in T or in a parallel link of T and let y ′ b e the first no de after x ′ that is in V ( H ). Th en x ′ → y ′ has pr op erties 2.8.1 and 2.8.3 and is a BG-path for H . • H = smo oth ( H ) . Then H consists only of real no des and s in ce H 6 = G , ther e is a no de in V ( G ) \ V ( H ) or an edge in E ( G ) \ E ( H ). A t first, assume that there is a no d e x ∈ V ( G ) \ V ( H ). Then, by the 2-connectedness of G and F an Lemma 2.1 w e can find a p ath P = y 1 → x → y 2 with n o other no des in H than y 1 and y 2 . F or P the p r op erties 2.8.1-2.8.3 hold, b ecause no link in H can h a ve in ner no d es. Let now V ( G ) = V ( H ) and e an edge in E ( G ) \ E ( H ). Th en e m ust b e a BG-path for H , since b oth end no des are real. In Theorem 3.1, non-basic op erations can only o ccur in the case H = smo oth ( H ) when a p ath throu gh a n o de of V ( G ) \ V ( H ) is c hosen. Although we cannot a v oid that, it is p ossible to obtain a basic construction b y augmenting the BG-operations w ith a fourth op eration (d). (d) connect a new n o de to three distinct no des 640 J. M. SCHMIDT Figure 4: A 3-connected graph ha ving a no de x of degree 3 with no inciden t edge b eing remo v able. Op eration (d) preserves 3-connectedness with Lemma 2.2 and is basic, b ecause eac h new edge ends on th e new no de. Wheneve r w e encount er a no de in V ( G ) \ V ( H ) in Theorem 3.1, w e know by the F an Lemma 2.1 and the 3-connectedness of G that there are three in ternally no de-disjoint p aths to real no des in H with all inner no d es b eing in V ( G ) \ V ( H ). Adding these p aths to H is called an exp and op eration and corresp onds to op eration (d ) in the smo othed graph. This giv es the follo wing result. Theorem 3.2. L et G b e a simple gr aph and let H b e a sub divi si on of a 3 -c onne cte d gr aph. Then G is 3 -c onne cte d and H ⊆ G ⇔ δ ( G ) ≥ 3 and ∃ c onstruction se quenc e f r om H to G using BG-p aths (3.1) ⇔ δ ( G ) ≥ 3 and ∃ b asic c onstruction se quenc e fr om H to G u si ng B G- p aths and the exp and op er ation (3.2) Pr o of. Let G b e 3-connected and H ⊆ G . Then δ ( G ) ≥ 3 holds and if H = G , th e desired construction sequences are empt y and exist. I f H ⊂ G , we can apply Theorem 3.1 iterativ ely with or without th e additional expand op eration and the construction sequences exist as w ell. F or th e su fficiency p art, b oth construction sequences imply H ⊆ G , since only paths are added to construct G . Additionally , G m ust b e 3-connected, as adding BG-paths to eac h S i preserve s S i +1 to b e a sub division of a 3-connected graph with Th eorem 2.4, and δ ( G ) ≥ 3 ensu res that the last sub division G of a 3-c onnected graph is 3-connected itself. 4. Represen t ations A straight -forw ard algorithm to compute Barnette and Gr ¨ unbaum’s constru ction se- quence of a 3-connected graph is to searc h iterativ ely for remo v able edges. But in con trast to the algorithm in Section 2.1 that computes con tractible edges, this ap p roac h only leads to an O ( n 3 ) algorithm. T h e reason for the additional factor of n is that not all n o des with degree 3 m ust h a ve an inciden t r emo v able edge (see Figure 4 for a count erexample on 9 no des) and we ha v e to try ev ery edge in the wo rst case. Computing BG-paths instead of BG-op erations allo ws us to obtain b etter runn ing times, but first w e need to kno w ho w exactly construction sequ ences can b e represen ted. An obvious representat ion of a construction s equ ence Q would b e to store the graph G 0 = smo oth ( H ) and in add ition ev ery BG-o p eration, which giv es the sequence G 0 , . . . , G z = G . Unfortun ately , the graphs G i are not necessarily su bgraphs of G i +1 , so w e ha v e to tak e care of relab eled edges when sp ecifying eac h op eration. Construction Sequences and Certifying 3-Connectedn ess 641 Figure 5: The case H 6 = smo oth ( H ). Dashed edges are in E ( G ) \ E ( H ), arr o w s depict the BG-path x ′ → y ′ . Whenev er an edge e is sub divided as p art of an op eration (b) or (c), we sp ecify it by its index in G i follo wed b y assigning new ind ices for the new degree-t w o n o de and one of the t w o new s ep arated edge parts in G i +1 . Th e other edge part kee ps the index of e . Similarly , on op erations (a) and (b), real endno des of the added edge are sp ecified b y their indices in G i . W e assign a new index for the added edge in G i +1 , to o. Finally , w e hav e to imp ose the constraint that G z is n ot just isomorphic bu t id entica l to G , meaning that no des and edges of G z and G are lab eled by exactly the same indices, since otherwise w e w ould ha v e to solv e the graph isomorphism p r oblem to c h eck that Q really constructs G . On the other hand, the id en tification of G i with a subgraph in G allo ws us to r epresen t Q without indexing issues: W e ju st store S 0 ⊂ G and the BG-paths C 0 , . . . , C z − 1 . Hence, w e can r epresen t eac h construction sequ ence Q of G in the follo wing t wo wa ys. • Edge r epr esentation : Represent Q by G 0 and a sequence of BG-op erations, along with sp ecifying n ew and old in dices for eac h op eration, su c h that G z and G are lab eled the same. • Path r epr esentation : Repr esen t Q by S 0 and BG-paths C 0 , . . . , C z − 1 . Both representati ons refer to the same s equ ence of graphs G 0 , . . . , G z and are of size θ ( m ), assu ming the un if orm cost mo d el. The next lemma states th at it d o es n ot matter whic h of the tw o representat ions we compute. Lemma 4.1. The e dge and p ath r epr esentations of a c onstruction se qu enc e Q c an b e tr ans- forme d into e ach other in O ( m ) time. Mor e over, the r e pr esentation c ompute d is a unique r epr esentation of Q . Pr o of. Omitted. 5. Certifying and T esting 3 -Connectedness in O ( n 2 ) W e use construction sequences in the path rep r esen tation as a certificat e for the 3- connectedness of graph s. This leads to a new, certifying metho d for testing graphs on b eing 642 J. M. SCHMIDT 3-connected. The total running time of th is method is O ( n 2 ), how ev er this is domin ated b y the time needed for fi n ding th e construction sequence and every improv emen t mad e there will automatically r esult in a faster 3-connectedness test. The input graph is a multig raph and do es n ot ha ve to b e biconnected nor connected. W e follo w the steps: • Apply prepro cessing of Nagamo c hi and Ibaraki to th e graph and get G in O ( n + m ) (This impro ves the total run ning time b y decreasing the n um b er of edges to O ( n ).) • T r y to compute a K 4 -sub d ivision S 0 in G and prescrib e it in O ( n ) – F ailure: Return a separation pair • T r y to compute a constru ction sequence fr om S 0 to G in O ( n 2 ) – Success: Return the constru ction sequence – F ailure: Return a separation pair Figure 6: Finding a K 4 - sub d ivision. Dashed edges can b e (empty) paths, arcs d epict bac kedges. The prepro cessing step preserv es the g raph to b e 3- connected or to b e not 3-connected. W e first describ e ho w to find a K 4 -sub d ivision by one Depth First Searc h (DFS), w hic h as a b yp r o duct eliminates self-lo ops and p ar- allel edges and sorts out graphs that are not connected or ha v e no d es with degree at m ost 2. Let a (resp. b ) b e the no de in the DFS-tree T that is visited firs t (resp. s econd). If G is 3-connected, th en a and b h a ve exactly one c h ild , otherwise they form a separation pair. W e c h o ose t w o ar- bitrary neigh b ors c and d of a th at are d ifferen t from b (see Figure 6). W.l.o.g., let d b e visited later b y the DFS than c . Let i 6 = b the least co mmon ancestor of c and d in T . As d 6 = i must hold, let j b e the child of i that is con tained in the path i → d in T . If G is 3-connected, w e can find a back edge e that starts on a no de z in the su btree ro oted at j and ends on an inner no de z ′ of a → i in time O ( n ). If e do es not exist, a and i form a separation pair, otherwise we ha v e f ou n d a K 4 -sub d ivision w ith real n o des a , i , z and z ′ . The paths connecting this real no des in T together with the three visited back edges constitute the 6 paths of the K 4 -sub d ivision. Once the K 4 -sub d ivision S 0 is foun d, we follo w the lines of T heorem 3.1 and try to construct th e path rep r esen tation C 0 , . . . , C z − 1 . If fa v ored, this can b e transformed to an edge repr esen tation in O ( m ) late r. W e assign an index f or ev ery link and store it on eac h of the inner no des of that link. Moreov er, we main tain p ointe rs for eac h lin k to its end no des. In case H 6 = smo oth ( H ) of Theorem 3.1 we pic k an arbitrary n o de x of degree t wo . L et T = a → b b e the link that con tains x and let W b e the set of nod es V ( H ) \ V ( T ) minus all no des in parallel links of T (see Figure 5). W e compu te the path P = x → y ′ b y temp orarily deleting a and b an d p erforming a DFS on x th at stops on the first n o de y ′ ∈ W . W e can c h ec k wh ether a no de lies in a parallel link of T in constan t time by comparin g the endn o des of its conta ining link with a and b . Th us, th e s u bpath x ′ → y ′ with x ′ b eing th e last no de con tained in T or in a p arallel link of T is a BG-path an d can b e found efficien tly . The links and their in dices can b e up d ated in O ( n ). Similarly , in case H = smo oth ( H ) we delete temp orarily all edges in E ( H ) and start a DFS on a no de x ∈ V ( H ) th at has an incident edge in the r emaining graph. The tra versal is stopp ed on th e first n o de y ∈ V ( H ) \ { x } . The path x → y is then the desired BG-path Construction Sequences and Certifying 3-Connectedn ess 643 (a) Either a or b has degree 2. (b) Both, a and b , ha ve degree 2. Figure 8: C ases where 2.8.2 fails when a ∈ N ( b ). and we conclude th at for 3-connected graphs the construction sequence can b e found in time O ( n 2 ). Otherwise, G is not 3-connected and n o construction sequen ce can exist with Theo- rem 3.2. In that case a DFS starting at no de x fails to find a new BG-path for some sub d ivision H ⊂ G . I f H 6 = smo oth ( H ), the endn o des of the link that contai ns x must form a separation pair. Oth erwise, H = smo oth ( H ) and x m ust b e a cut v ertex. Th us, if G is not 3-connected, the algorithm return s alw ays a separation p air or cut vertex. If G is simple, the constru ction sequence can b e transf ormed to the basic construction sequence (3.2) with the follo wing Lemma. Lemma 5.1. F or simple gr aphs G , the c onstruction se quenc es (3.1) and (3.2) c an b e tr ans- forme d into e ach other in O ( m ) . Pr o of. Omitted. Theorem 5.2. The c onstruction se quenc es (3.1) and (3.2 ) c an b e c ompute d in O ( n 2 ) and establish a c ertifying 3 -c onne cte dness test with the same running time. 5.1. V erifying the C onstruction Se quence Figure 7: No e xpand op era- tion can b e formed. It is essenti al for a certificate that it can b e easily v al- idated. W e could do this by transform ing the path r epre- sen tation to the edge representati on using Lemma 4.1 and c h ec king the v alidit y of the BG-o p erations by comparing indices, but there is a more direct w a y . First, it can b e c h ec ked in lin ear time that all BG-paths C 0 , . . . , C z − 1 are paths in G and that these paths partition E ( G ) \ E ( S 0 ). W e try to remov e the BG- paths C z − 1 , . . . , C 0 from G in that order (i. e. , w e delete the paths f ollo we d b y smo othing its endn o des). If the certificate is v alid, this is well defined as all remo v ed BG-paths are then edges. On the other h and w e can detect longer BG- paths | C i | ≥ 2 b efore their r emov al, in which case the certificate is not v alid, since then th e inner no des of C i are not attac hed to BG-paths C j , j > i . W e verify that ev ery remo v ed C i = ab corresp ond s to a BG-op eration by using Defini- tion 2.8 of BG-paths, and start w ith c h ec king that a and b lie in our cur r en t s ubgraph f or condition 2.8.1. 644 J. M. SCHMIDT Conditions 2.8.2 and 2.8.3 can n ow b e chec k ed in constant time: Consider the situation immediately after the deletion of ab , but b efore smo othing a and b . Th en all links in our subgraph are single edges, except p ossibly the ones con taining a and b as inn er no des. Therefore, 2.8.2 is n ot met for C i if a is a neighbor of b and at least one of the no des a and b has degree tw o (see Figures 8 for p ossible configur ations). Condition 2.8.3 is not met if N ( a ) = N ( b ) and b oth a and b ha ve degree t wo. Both conditions can b e easily c hec k ed in constan t time. Note that encount ering pr op er BG-paths C z − 1 , . . . , C i do es not necessarily imply that the current subgraph is 3-connected, since false BG-paths C j , j < i , can exist. 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