Recognizing Cartesian products in linear time

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1 Dscrete Mathematcs 307 (2007) Recognzng Cartesan products n lnear tme Wlfred Imrch a, Iztok Petern b a Char of Appled Mathematcs, Montanunverstät Leoben, A-8700 Leoben, Austra b Faculty of Electrcal Engneerng and Computer Scence, Unversty of Marbor, Smetanova ulca 17, 2000 Marbor, Slovena Receved 12 September 2003; receved n revsed form 17 February 2005; accepted 26 September 2005 Avalable onlne 1 September 2006 Abstract We present an algorthm that determnes the prme factors of connected graphs wth respect to the Cartesan product n lnear tme and space. Ths mproves a result of Aurenhammer et al. [Cartesan graph factorzaton at logarthmc cost per edge, Comput. Complexty 2 (1992) ], who compute the prme factors n O(m log n) tme, where m denotes the number of vertces of G and n the number of edges. Our algorthm s conceptually smpler. It gans ts effcency by the ntroducton of edge-labellngs Elsever B.V. All rghts reserved. Keywords: Cartesan product graphs; Lnear algorthm; Decomposton 1. Introducton Cartesan products are common n graph theory. Typcal examples are hypercubes, Hammng graphs, and grd graphs. Hypercubes are powers of K 2, Hammng graphs products of complete graphs, and grd graphs products of paths. These products and ther sometrc subgraphs have numerous applcatons n dverse areas, such as Computer Scence, Mathematcal Chemstry and Bology. The Cartesan product also has unque algebrac, structural and metrc propertes. They were nvestgated n the 1960s by Sabduss [7] and Vzng [8]. One of these propertes s the representaton of graphs as the Cartesan product of prme graphs, where a graph s called prme f t cannot be presented as the product of two nontrval graphs, that s, as the product of two graphs wth at least two vertces. Independently Sabduss and Vzng showed that every connected fnte graph has a prme factor decomposton wth respect to the Cartesan product that s unque up to the order and somorphsms of the factors. (For dsconnected graphs the factorzaton s not unque.) Sabduss also studed Cartesan products of fnte or nfnte graphs wth nfntely many factors and Vzng the domnaton number. A conjecture of Vzng [9] from that tme about the domnaton number of Cartesan products s stll open. 1 Wth the advent of complexty theory n the 1970s the queston arose whether one could fnd the prme factorzaton of connected graphs n polynomal tme. The frst postve answer was gven n 1985 by Fegenbaum et al. [4], who presented an algorthm of complexty O(n 4.5 ), where n denotes the number of vertces of the nvestgated graph. Ther work extends a method of Sabduss [7]. Wnkler [10] ndependently found an entrely dfferent algorthm of complexty E-mal addresses: mrch@unleoben.ac.at (W. Imrch), ztok.petern@un-mb.s (I. Petern). 1 Vzng conjectures that the domnaton number of the Cartesan product of two graphs s bounded from below by the product of the domnaton numbers of the factors X/$ - see front matter 2006 Elsever B.V. All rghts reserved. do: /j.dsc

2 W. Imrch, I. Petern / Dscrete Mathematcs 307 (2007) O(n 4 ) and Feder [3] contnued wth an algorthm that requres O(mn) tme and O(m) space. Ths was further mproved to O(m log n) tme and O(m) space by Aurenhammer et al. [2]. The algorthm presented here s lnear n tme and space and conceptually smpler. For a gven graph G t computes the partton of the edge set of G that s assocated wth the prme factor decomposton of G. Ths edge partton s an ntrnsc, metrc property of G. We only have to fnd t, we do not mpose any addtonal structure on G by decomposng t nto prme factors. No coordnatzaton of the vertces s needed. In order to prove ts correctness, we rely on result about the structure of Cartesan products as presented n [6]. The dea behnd our algorthm s smple: gven a connected graph G, select a vertex v 0 of mnmum degree, lst the edges n BFS order wth respect to v 0. Assume that every edge ncdent wth v 0 s n a dfferent factor. Scan the edges n BFS order and use ths nformaton to determne to whch factor the other edges belong. Ths wll be called the colorng and labellng algorthm. If t fals for an edge, then there are too many factors. Merge them as needed. Then check whether the condtons for Cartesan products are satsfed. Ths s the consstency check. If t fals, then there are too many factors. Merge as requred. The algorthm ends when the last edge has been successfully processed. 2 The algorthm s lnear, 3 that s, lnear n the sze of the nput. For a connected graph ths s equvalent to sayng t s lnear n the number m of edges f the graph s gven by ts adjacency lst. Snce every edge has to be processed, every colorng and labellng operaton as well as every consstency check must be effected n constant tme. For the merge operatons we observe that there cannot be more factors than the mnmum degree d 0, hence at most d 0 merge operatons are necessary. For each one of them we can use O(n) tme, where n s the number of vertces, because nd 0 2m. Ths s only possble wth a carefully chosen data structure, whch also has to satsfy the addtonal restrcton of O(m) space. The paper begns wth three fundamental lemmas about the Cartesan product. Each of these lemmas s the bass of one of the man parts of the algorthm. The Square Lemma s essental for the colorng and labellng algorthms, the Isomorphsm Lemma for the consstency check and the Refnement Lemma for the correctness of the merge operatons. 2. Prelmnares The Cartesan product G H of the graphs G = (V (G), E(G)) and H = (V (H ), E(H )) s a graph wth vertex set V (G) V(H), where the vertces (a, x) and (b, y) are adjacent f ab E(G) and x = y,orfa = b and xy E(H). The Cartesan product s assocatve, commutatve, and has the one vertex graph K 1 as a unt. By the assocatvty we can wrte G 1 G k for a product G of graphs G 1,...,G k and can label the vertces of G by the set of all k-tuples (v 1,v 2,...,v k ), where v G for 1 k. Ifv s labelled by (v 1,v 2,...,v k ) we set p v = v and call v the th coordnate of v. The mappng p projects G onto G. If we restrct p to the subgraph nduced by all vertces that dffer from a gven vertex w only n the th coordnate, t clearly becomes an somorphsm. Ths subgraph s known as the G -layer through w and denoted by G w. As an example, consder the graph G of Fg. 1. Consdered as C 5 C 4 the layers are fve- and four-cycles, consdered as C 5 K 2 K 2 the layers are fve-cycles and sets of parallel edges. For the dstance d G (u, v) between two vertces u, v V (G) we have d G (u, v) = k d G (p u, p v). (1) =1 Ths mmedately mples that the layers of a product are convex subgraphs, that s, every shortest path between two vertces of one and the same G w s already contaned n G w. In the sequel t wll be convenent to select a root v 0 G and to dentfy every G wth G v 0. The layers through v 0 are then called unt layers. Wth ths termnology every vertex v G s equal to p v. Moreover, v 0 s contaned n all factors G, but otherwse the sets V(G )\{v 0 } are mutually dsjont. The reader mght lke to vsualze the unt-layers of the graph G n Fg. 1 both as a product C 5 C 4 and C 5 K 2 K 2. 2 See also the short summary of the algorthm at the end of the paper. 3 In Secton 3 we also descrbe a techncally much smpler algorthm of complexty O(m 2 ).

3 474 W. Imrch, I. Petern / Dscrete Mathematcs 307 (2007) Fg. 1. G = C 5 K 2 K 2 = C 5 C 4. The premage p 1 (v) of any vertex v G s a set of vertces n G that nduces a subgraph somorphc to G = k = G k. Any such graph s a G -layer of G G. Furthermore the subgraph nduced by p 1 (u) p 1 (v), where uv E(G ), s somorphc to K 2 G. We call t the tower wth base uv. The K 2 -layers of ths product are a matchng between p 1 (u) and p 1 (v). These matchng edges nduce an somorphsm between the subgraphs of G nduced by p 1 (u) and p 1 (v). We call the set of these edges the premage of uv E(G ). (The tower wth base of v 0 v n Fg. 1 s easly vsualzed as K 2 C 5 for the representaton of G as C 5 C 4. For the representaton of G as C 5 K 2 K 2 the tower s K 2 (C 5 K 2 ), that s, the graph G tself.) We now defne an edge-colorng that reflects the layer structure of G. We frst observe that the coordnate vectors 4 of two adjacent vertces u, v dffer n exactly one place. Let ths place be. Then uv G u and we say that uv has color, n symbols c(uv) =. We call ths a proper product colorng of G. (The graph G of Fg. 1 admts fve proper product colorngs. One wth three colors for the representatons C 5 A B, where A B K 2, three wth two colors for the decompostons C 5 (A B), (C 5 A) B, (C 5 B) A, and one wth one and the same color for all edges.) Every edge s contaned n exactly one layer; the edge sets of the layers of G partton the edge-set of G. The edges of color n the product G = G 1 G k form a spannng subgraph of G and every connected component of ths subgraph s a G -layer. The projectons p and the somorphsm propertes of the layers of G allow the characterzaton of Cartesan products, as we shall llustrate below. It s easy to see that the restrcton of p to G w s an somorphsm from G w onto G. If there s an edge between two G -layers G u and Gv we say they are adjacent. Clearly the edges between any two adjacent G -layers G u and Gv nduce an somorphsm (and a matchng) between them. It s (p v) 1 p u, where px j denotes the restrcton of p to G x. Suppose e 1 e 2...e l s a path P that connects the G -layer G u wth Gw, and that no e has color. Then concatenaton of the somorphsms nduced by the e j yelds an somorphsm of G u onto G w. (Note that p (P ) s a sngle vertex.) Ths somorphsm s unquely determned and s, analogously to the above, just (p w) 1 p u. Ths s a consequence of the fact that every maxmal connected subgraph of G that contans no edges of color, where s arbtrarly chosen, meets every G layer n exactly one vertex. Now the lemma. Lemma 2.1 (Isomorphsm Lemma). Let G = (V, E) be a connected graph and E 1,E 2,...,E k a partton of E. Suppose that every connected component of (V, j = E j ) meets every connected component of (V, E ) n exactly 4 Recall that the coordnate vectors are just labels of length k that satsfy several propertes, see above.

4 W. Imrch, I. Petern / Dscrete Mathematcs 307 (2007) Fg. 2. The square property. one vertex and that the edge set between any two components of (V, E ) nduces an somorphsm between them f t s nonempty. Then G = G, where the G are arbtrary components of the graphs (V, E ). A graph satsfes the somorphsm property f t satsfes the assumptons of the Isomorphsm Lemma. Let two components C and D of (V, E ) be adjacent. To verfy that the edges between adjacent components C and D of (V, E ) nduce an somorphsm between them t clearly suffces to check that every edge of C and D s contaned n exactly one square consstng of one edge n C, one n D and two edges between C and D. Ths gves rse to the Square Lemma. Lemma 2.2 (Square Lemma). Let G be a properly colored Cartesan product. If e and f are ncdent edges of dfferent colors, then there exsts exactly one square wthout dagonals that contans e and f. It s clear what we mean by the square property, see Fg. 2. Suppose the edges of a graph G are colored n such a way that the square property s satsfed. Then we can nfer rather strong assertons about the colorng of certan nduced subgraphs. For example, all edges n a trangle have the same color and all edges n a square wth at least one dagonal have the same color. Also, all edges n a K 2,3 are of the same color, because any two edges ncdent n a vertex of degree two are contaned n two squares wthout dagonals, and not just one. Furthermore any two opposte edges of a square have the same color and, most mportantly, any two ncdent edges uv, vw have the same color f v s the only common neghbor of u and w. We conclude ths secton wth a formulaton of the unque prme factorzaton property of connected graphs as we shall use t n ths paper. Before statng t we wsh to remark that we do not always dstngush between a partton of a set and the assocated edge-colorng or the equvalence relaton whose equvalence classes are just the sets n the gven partton. The equvalence relatons nherted from a product decomposton are also called product relatons. Lemma 2.3 (Refnement Lemma). All product relatons wth respect to the prme factor decompostons of a connected graph G are dentcal or fner than any other product relaton of G. For a proof we refer to [6]. Note that the product relaton, resp. product colorng, that corresponds to the decomposton C 5 K 2 K 2 of the graph G of Fg. 1 s fner than all the other product relatons of G. Thus C 5 K 2 K 2 s the prme factorzaton of G. Of course t s not dffcult to see drectly. Clearly ths holds n general the fnest colorng that satsfes the somorphsm property gves rse to the unque fnest product relaton and therefore, to the prme factorzaton of connected graphs. It s our am to fnd ths colorng n lnear tme and space. 3. A drect algorthm for the fnest product relaton The am of ths secton s to demonstrate that the fnest product relaton s an ntrnsc relaton of connected graphs that depends only on the metrc. It also shows that there s a straghtforward O(m 2 ) algorthm to compute the fnest

5 476 W. Imrch, I. Petern / Dscrete Mathematcs 307 (2007) product relaton and thus the prme factorzaton of a connected graph. The results shed some lght on the underlyng deas of the paper but are not needed n the sequel, so ths secton can be skpped. We begn wth the defnton of two relatons Θ and τ on the edge set E(G) of a connected graph G. Two edges e = xy and f = uv are n the relaton Θ f d(x,u) + d(y,v) = d(x,v) + d(y,u), (2) and they are n the relaton τ f y = u s the only common neghbor of x and v. Let σ denote the transtve closure of the unon of Θ wth τ, σ = (Θ τ). Theorem 3.1 (Feder [3]). Let G be a fnte connected graph. Then (Θ τ) s the fnest product relaton of G. For a proof see [3] or [6]. To see how the theorem works, consder the graph G of Fg. 1. Any two adjacent edges n one of the pentagons are n the relaton τ, and thus any two nonadjacent edges of a pentagon n the relaton τ, that s, n the transtve closure of τ. Moreover, any two parallel edges (n the fgure) are n the relaton Θ. Thus, any two edges n the unon of the pentagons are n the relaton σ. Any two horzontal edges are n the relaton Θ as well as any two of the remanng edges. Ths gves a partton of E(G). It s readly checked that no two edges n dfferent sets of ths partton are n the relaton Θ or τ. We have thus determned the relaton σ G. Clearly t gves rse to the factorzaton C 5 K 2 K 2. Lemma 3.2. The prme factorzaton of a connected graph on m edges can be determned wthn tme and space complexty O(m 2 ). Proof. It suffces to show that σ can be determned wthn tme and space complexty O(m 2 ). To see ths, we begn wth Θ and observe that there are m 2 pars of edges n G. For every such par we have to fnd four dstances, and check whether Eq. (2) s satsfed. Ths can be done n constant tme wth the ad of the dstance matrx, whch can be determned n O(mn) tme. To fnd τ t suffces to consder all vertces u and check for any other vertex v whether t s adjacent to u or any of ts neghbors. Thus, there are n (d(u) + 1) adjacences to be checked for every vertex u. Snce n(d(u) + 1) = n (d(u) + 1) = O(nm), u u the tme complexty for determnng τ s O(nm). Thus, the at most m 2 pars of edges n that are n the relatons Θ or τ can be found wthn tme complexty O(m 2 ). Hence the transtve closure of Θ τ can also be found wthn O(m 2 ) tme. 5 Clearly ths straghtforward algorthm also requres O(m 2 ) space. Theorem 3.1 s a very specal result, because t tells us two thngs: t makes a deep asserton about the structure of Cartesan products and smultaneously yelds a straghtforward polynomal algorthm to fnd the prme factorzaton of connected graphs. Interestngly, ths theorem also holds for nfnte connected graphs, see [5]. It should be noted that Feder [3] mproved the complexty of ths approach by showng that Θ can be determned n O(mn) tme and space. As (Θ τ) = (Θ τ) t s not necessary to determne Θ. Ths also mples that Θ cannot contan more than O(mn) pars of edges. Snce ths also holds for τ both the space and tme complexty of the prme factorzaton reduce to O(mn). In the next secton we develop a dfferent approach for fndng the product colorng of a connected graph. 5 Consder a graph H wth V(H)= E(G), where any two edges e, f are adjacent n H f they are n at least one of the relatons Θ or τ. Then the equvalence classes of σ correspond to the connected components of H and can be found wthn tme complexty O( E(H) ) = O(m 2 ).

6 4. Factorzaton wth addtonal nformaton W. Imrch, I. Petern / Dscrete Mathematcs 307 (2007) Clearly every graph s a Cartesan product albet a trval one n most cases. Our task s to fnd the decomposton. Here we fnd t wth some addtonal nformaton. We show how to fnd the product colorng of a graph, provded that the colors of the edges ncdent wth a vertex v 0 are known. The only other data of G that we requre s the adjacency lst. The mportance of the followng proposton s the method of provng t: one begns wth the colorng of the edges ncdent wth a root vertex v 0 and extends t to all edges of the graph n BFS order. The complexty of ths approach s O(mn). Proposton 4.1. Let G = G 1 G 2 G k be a connected graph. Suppose the colors of the product colorng of G wth respect to the gven decomposton are known for the edges ncdent wth a root vertex v 0. Then the product colorng can be recovered n BFS order wth respect to the root v 0 n O(mn) tme and O(n 2 ) space. Proof. We begn by arrangng the vertces of G n BFS order wth respect to the root v 0 and denote the dstance levels wth respect to ths orderng by L, = 1,...,r. In other words, L ={v d G (v 0,v)= }. Furthermore we partton the set of edges ncdent wth every vertex u nto three sets: the set of down-edges, cross-edges and up-edges. Note that an up-edge uv of u s a down-edge of v. It s thus convenent to consder our edges as ordered pars of vertces, 6 such that the statement xy s a down-edge means that xy s a down-edge wth respect to x. For cross edges the statement the cross edge xy wll mean the cross edge xy wth respect to the vertex x. We color the cross edges of L 1 frst. Ths s easy, because every trangle s monochromatc. For every cross-edge uv n L 1 we only have to set c(uv) = c(uv 0 ). Then we proceed by nducton. We assume that we have colored every edge up to the cross-edges of L and contnue wth the down-edges and then the cross-edges of L +1. Here we do not have to consder the up-edges of L separately, because they are also down-edges of L +1. To color the down-edges of L +1 we scan the vertces u L +1 n BFS order and fx a down-edge uv. 1. Suppose ths s the only down-edge of u. Snce v L, 1, there exsts a down-edge vx. Clearly v s the only common neghbor of u and x. Thus c(uv) = c(vx). 2. The other possblty s that there are other down-edges. For every such down-edge uw we look for a vertex x that s adjacent to v and w. If such a vertex exsts, then c(uv) = c(wx) and c(uw) = c(vx), no matter whether ux and wx have the same or dfferent colors. If no such x exsts, then we consder any down-neghbor x of v. As before, v s the only common neghbor of u and x and so c(uv) = c(vx). Analogously we color uw n ths case. For the cross-edges of L +1 we scan the vertces of L +1 agan. For every vertex u that s an endpont of a cross-edge we select an arbtrary cross-edge uv and a down-edge uw. As before we look for common neghbors x of v and w. If such an x exsts, then c(uv) = c(wx), otherwse c(uv) = c(uw). To fnd the G we observe that t suffces to fnd the G v 0. Snce they are convex, a vertex v s n V(G v 0 ) f and only f all of ts down-neghbors have color. A scan of V (G) readly produces these vertces, and the adjacency lsts of the G v 0 are just the -chromatc sublsts of the adjacency lst of G, restrcted to V(G v 0 ). In Secton 2 we called the G v 0 unt-layers. We wll call ther vertces unt-layer vertces henceforth. Clearly all vertces of L 1 are unt-layer vertces. All down-neghbors and cross-neghbors of a unt-layer vertex u G v 0 are also n G v 0. Ths s a consequence of the convexty of layers n a product. (See also Lemma 7.29 on p. 235 n [6].) For an estmate of the complexty of our procedure we note that we fx a down-neghbor v of every vertex u L, >0, and then scan all cross- and down-edges of u, so we have a total of at most 2m steps to perform, where m denotes the number of edges of G. For every par of the already determned edges uv and uw we then search for common neghbors of v and w. We have fewer than n = V (G) possble choces for x. If we work wth the adjacency matrx of G the checks whether x s adjacent to u or w can be performed n constant tme. Thus, we arrve at an overall complexty of O(mn) tme and O(n 2 ) space for ths unrefned approach. 6 If we consder every edge [u, v] as a set consstng of the two arcs (u, v) and (v, u) we obtan a partton of the set of arcs of G.

7 478 W. Imrch, I. Petern / Dscrete Mathematcs 307 (2007) Coordnatzng a product In our algorthm we do not need the coordnatzaton of the vertces. We wsh to show here that one can obtan the coordnates of a Cartesan product n lnear tme and space once the colors of the edges are known. Theorem 5.1. Let G = G 1 G 2 G k be a connected graph. Suppose the product colorng of G wth respect to the gven decomposton s known. Then the product can be coordnatzed n O(m) tme and space. Proof. Agan we begn wth a BFS orderng of the vertces. For the coordnatzaton then need a coordnate vector of length k for every vertex, where k s the number of factors. Clearly k s bounded by the mnmum degree d 0 of G. Snce nd 0 2m the total space needed for these vectors s O(m). By defnton the th coordnate of a vertex u s p (u). If we dentfy every G wth the unt-layer G v 0 and the vertces of G wth ther BFS-numbers, then the th coordnate of u s the BFS-number of p (u). As shown n [6] we can then coordnatze the vertces of G as follows: Begn the BFS-numberng wth 0 and let the coordnate vector of v 0 consst of k zeros. Then scan all other vertces u of G n BFS order. If u s a unt-layer vertex n G v 0 set the th place of u equal to the BFS-number of u and all other places to zero. If u s not a unt-layer vertex there must be down-edges uv and uw of dfferent colors. Set u = max(v,w ) for 1 k. Clearly the tme complexty for ths procedure s O(m). 6. Labellng the edges of a product In ths secton we refne the colorng of the edges and call t labellng. All edges n the premage of a unt layer edge wll receve the same label. We show that products can be labelled, and thus also colored, n lnear tme. Ths labellng allows to tell the poston of every edge wth respect to the gven product decomposton n constant tme. In the next one and a half pages we descrbe the data structure used for storng the labels. Actually, the label wll turn out to be the poston of an edge n an array of edges wth the same endpont and color. Note that we used the coordnate vectors for the characterzaton of the poston of vertces n the product, the total length of these vectors beng O(m). For the poston of an edge uv n the product we use the ntal vertex u, the color of uv and the projecton p (uv), that s, p up v. Ths edge s the base of uv. Itsntheth unt layer G v 0, has the same color as uv, and, because of Eq. (1), s a down-, resp., cross-, or up-edge, f and only f uv s a down-, resp., cross-, or up-edge. Below we descrbe how to store nformaton about the base effcently. The BFS-orderng of Secton 4 parttons the set of edges ncdent wth every vertex nto arrays of down-, cross-, and up-edges. Every such array s further parttoned nto subarrays of edges of the same color. We use the poston j of p (uv) n ts sublst to be able to locate p (uv) fast and call n(uv)=j the name of uv. In general n(uv) wll be dfferent from n(vu). The par c(uv), n(uv) s then the label of uv, n symbols l(uv). Together wth p c(uv) u t descrbes the poston of uv n the product. The poston of the ntal edge of every color n such an array of down-, cross-, and up-edges s stored n a vector of length d(v 0 ), thus the jth element of any such subarray of edges of the same color can be accessed n constant tme. Clearly the labellng s well defned. It depends on the orderng of the down- and cross-edges of every unt-layer vertex (all down- and cross-edges of a unt-layer vertex have the same color) and on the orderng of the up-edges of color n the lsts of up-edges of color for the vertces of G v 0. (Up-edges of other colors of the vertces n G v 0 dfferent from v 0 are not unt-layer edges.) The orderng of the other monochromatc sublsts of the lsts of down-, cross-, and up-edges does not effect the labellng. We shall reorder the monochromatc sublsts of edges that are not unt-layer edges accordng to ther names. It s convenent to generate an edge-lst that tells of every edge uv ts orgn u, ts termnus v, whether t s a downcross- or up-edge as well as ts color and name. Ths way we can fnd the poston of the edge n ts monochromatc subarray n constant tme once ts number s known.

8 W. Imrch, I. Petern / Dscrete Mathematcs 307 (2007) We also modfy the adjacency matrx by nsertng the name of the edge uv n poston (u, v) n order to obtan the labellng of uv n constant tme f the adjacency of u and v s checked. Our data structure thus conssts of orgnal adjacency lst of the gven graph, the modfed edge lst, and the modfed adjacency matrx. Moreover the vertces are now arranged n BFS order, every vertex has a BFS-number and a BFSlevel, an optonal coordnate vector of length at most d(v 0 ), and s assocated wth three arrays: the arrays of down-, cross-, and up-edges. Every array s subdvded nto subarrays of monochromatc edges, where the postons of the ntal edge n the subarray are stored n a vector of length d(v 0 ). Clearly the tme needed to buld the data structure wthout the subdvson of these arrays s lnear n m. We have to show that the subdvson can also be effected n lnear tme. Wth the excepton of the adjacency matrx, the space needed s lnear n m too. It s the topc of the next secton to show that we do not need the full adjacency matrx and that we can make due wth constructng sngle lnes of the adjacency matrx a constant number of tmes. Theorem 6.1. Let G = G 1 G 2 G k be a connected graph. Suppose the colors of the product colorng of G wth respect to the gven decomposton are known for the edges ncdent wth a vertex v 0 of mnmum degree. Then the edges of G can be labelled n O(m) tme. Proof. We have to descrbe a lnear labellng algorthm. The orgnally colored edges can easly be labelled. Furthermore, snce every edge n L 0 L 1 s a unt-layer edge the arrays of down- and cross-edges of L 1 need not be parttoned. The colorng of the cross-edges of L 1 poses no dffculty, every such edge can be colored n constant tme, the names are assgned as the postons n the respectve sub arrays of up-, down-, or cross-edges. Suppose we have already labelled all edges up to level L such that we can access edges ncdent wth a gven vertex and label n constant tme. We scan all vertces of L +1 and treat the down edges frst. Let u L Vertces of down-degree one. If u has down-degree one, then the colorng procedure of Secton 4 for the only down-edge of u can be executed n constant tme. Its name s Fndng a pvot square for vertces of down-degree larger than one Frst run. Let uv be the frst down-edge of u and vx a down-edge of v. We now check for common vertces of u and x. Ths can be done by scannng the down-neghbors of u and checkng va the adjacency matrx whether they are also neghbors of x. (We use the lne of x.) The tme complexty for fndng such a jont neghbor (or to fnd out that none exsts) s O(d(u)). If no such neghbor exsts, then all down-edges of u and vx have the same color. It s trval to label all down-edges of u n ths case snce u s a unt-layer vertex and no partton of the array of down-edges s necessary. Suppose now that there exsts a common neghbor w of u and x. If vx and wx have dfferent colors, then we set l(uv) = l(wx) and l(uw) = l(vx). We can also easly fnd the postons of the ntal edges of every color n the subarrays of the array of down-edges of u. We only have to observe that the length of a subarray of any color = c(uv) has the same length n arrays of down-edges of u and v and that the length of the subarray of color c(uv) s the same for the down-edges of u and w. The labellng s effected n constant tme, the ntal edges of every subarray can be found n O(d(v 0 )) tme. We call uvxw a pvot square and use t to label the other down-edges of u, but before that we have to treat the case c(vx) = c(wx) Second run. If vx and wx have the same color, then uv and uw also have that color. If all other down-edges of v are of the same color, then u s a unt-layer vertex, a case that has been treated already. So, let vx be a down-edge wth c(uv) = c(vx ).There must be a common neghbor w of u and x, we can fnd t by scannng the down-neghbors of u and checkng the adjacency va the adjacency matrx. (We use the lne of x.) Now set l(uv) = l(w x ) and l(uw ) = l(vx ). The tme complexty for fndng w and x s O(d(u)). Agan the ntal edges of the subarrays of the array of down-edges of u can be found n O(d(v 0 )) tme. In ths case uvx w s a pvot square. 3. Labellng all down edges of u wth a pvot square. To label the other down-edges of u we use the pvot squares uvxw, resp., uvx w. For smplcty we rename x and w nto x and w for that purpose. Thus, let us scan all down edges of u and let uy be such a down-edge. Consder the down edge yz wth l(yz) = l(uv). Va the adjacency

9 480 W. Imrch, I. Petern / Dscrete Mathematcs 307 (2007) matrx we check whether z and v are adjacent. (We use the lne of v.) If they are, then l(uy) = l(vz). Otherwse c(uy) = c(uv), but then c(uw) = c(uy) and we can label uy by means of uw. (Usng the lne of w.) Clearly the tme complexty for every vertex y s constant, thus the total complexty for ths step s O(d(u)). Hence the complexty of labellng the down-edges of the vertces n L +1 s O d(u). u L +1 Smlarly one can show that the cross-edges of L +1 and the up-edges of L can be labelled wthn the same tme complexty. 7. Labellng n lnear tme and space Up to now we have ndscrmnately used the adjacency matrx whenever we wshed to check n constant tme whether two gven vertces were adjacent. Unfortunately t requres n 2 space, where n denotes the number of vertces of the graph. It exceeds our aspred space complexty and, even worse, ntalzng t would destroy the tme complexty. Luckly we need only one lne of the adjacency matrx at a tme, whch saves the space complexty. Concernng ntalzaton we note that t s possble to effectvely obtan the adjacency matrx for a graph G n O(m) tme, despte the fact that O(n 2 ) storage s requred. Ths can be accomplshed by a trck frst publshed by Aho et al. [1]. Applyng ther method to the lnes of the adjacency matrx we frst observe that the tme needed to generate the lne for a vertex x s proportonal to d(x). Snce x V (G) d(x) = 2m we stay wthn the aspred tme lmt, unless we repeatedly generated lnes of vertces wth hgh degree. Care has to be taken to avod ths, we wll show below how ths s acheved for the labellng of the down-edges. Theorem 7.1. Let G = G 1 G 2 G k be a connected graph. Suppose the colors of the product colorng of G wth respect to the gven decomposton are known for the edges ncdent wth a vertex v 0 of mnmum degree. Then G can be labelled n lnear tme and space. Proof. By Theorem 6.1 t suffces to show that the total space requrement of the labellng algorthm can be reduced to O(m) wthout ncreasng the tme complexty. Step 1 n the labellng algorthm for the down-edges of level L +1 for the vertces of down-degree one poses no problem. Step 2 s the search for a pvot square. Two runs may be necessary. In the frst run an arbtrary down neghbor x of an arbtrary down-neghbor v of the vertex u of L +1 that s beng consdered s needed for neghborhood-checkng. Let us denote x by x u. The down-neghbor x u of dstance two of another vertex u of L +1 may be dentcal wth x u. The dea s to determne all such x u before the lne of x u n the adjacency matrx s generated and to execute the frst run for all u wth x u = x u. The second run s treated just the same. In Step 3 two such procedures wll be necessary, ths tme for vertces n level L. For the cross-edges and the up-edges smlar procedures apply, but now thngs are slghtly easer snce the downedges are already labelled and snce the colors of the up-edges are already known. In ether case two such procedures may be necessary, yeldng a total of eght. Thus, no lne of the adjacency matrx wll have to be generated more than eght tmes. Snce the labellng s a refnement of the colorng, G can also be colored n lnear tme and space under the assumptons of the theorem. Moreover, by Theorem 5.1, t can also be coordnatzed wthn the same tme and space complexty. Note that the man lemma needed here was the Square Lemma. 8. Consstency check The precedng operatons worked under the assumpton that the gven graph was a product and that we knew the colors of the edges ncdent wth a gven vertex. If we are gven a colorng and asked to check whether t s a product

10 W. Imrch, I. Petern / Dscrete Mathematcs 307 (2007) colorng t mght be temptng to run the labellng algorthm wth the nformaton at one vertex and to check whether the computed colorng concdes wth the gven one. Of course, f t does not, or f the labellng algorthm breaks down, for example n Secton 6, Step 2.1, where t says there must be... then the gven colorng s not a product colorng. However, t s qute possble that the labellng algorthm goes through, even f these assumptons are not satsfed, for example n a cube wth a mssng edge or a mssng vertex. So we need a more effectve way of testng the valdty of the assumptons. We turn to the Isomorphsm Lemma for ths purpose. Isomorphsms of graphs are bjectons of the vertex-sets that preserve ncdence. Usng the BFS-orderng of our graphs we check nductvely whether the restrctons of these mappngs up to BFS level L satsfy the all the assumptons and contnue wth the next level thereafter. Proposton 8.1. Suppose that the somorphsm propertes hold up to level L. Then one can check n tme proportonal to the sum of numbers of down- and cross-edges of L +1 and the up-edges of L whether these propertes also hold up to L +1. Proof. We proceed n two steps. 1. For every vertex u L +1 that s not a unt-layer vertex we pck two down-edges uv and uw of dfferent color (we could use the pvot square for ths purpose) and check whether the down- and cross-edges of u and v that have a color dfferent from c(uv) are n one one correspondence. Ths means that for every down- or cross-edge uz of color = c(uv) there exsts exactly one edge vz such that l(uz) = l(vz ) and vce versa. Furthermore, zz must be an edge wth l(uv) = l(zz ). 2. For the up-edges of vertces n L we proceed smlarly. We scan all vertces u of L.Ifus not a unt-layer vertex, we choose a pvot square uvxw and test the up-edges of u aganst those of v and w. If u s a unt-layer vertex, say u G v 0, we select an arbtrary down-neghbor v of u and check the up-edges of u and v aganst each other that have a color dfferent from. Snce every sngle check can be performed n constant tme because of our edge-labellng the total tme needed for every vertex u s proportonal to the degree of u. Thus, the tme complexty for ths procedure remans lnear. Ths way we check the somorphsms between the layers G u and G v, resp., Gw. The others need not be checked. To see ths, consder an arbtrary neghbor a of u. Suppose t s a down-neghbor and that c(ua) = c(uv). We wsh to check the somorphsm between G u and G a nduced by the edges between them, where = c(uv). We have to show that for any down- or cross-edge ab of color = c(ua) there s exactly one edge uc wth the same label as ab such that b and c are adjacent and l(ua) = l(cb). Vce versa, to every down- or cross-edge uc of color = c(ua) there s exactly one edge ab wth the same label as uc such that b and c are adjacent and l(ua) = l(cb). We treat the frst case. We have already checked the exstence of an edge aa wth the same label as uv. Bythe nducton hypothess all somorphsms up to level L have been checked. Hence, we can use the Square Lemma. We complete {a,a,b} to a square abb a, then {v, a,b } to a square va b c and fnally {b, b,c } to a square bb c c. The edge vc has the same label as ab and cc the same label as uv. By the somorphsm check between u and v the vertces u and c are adjacent and uc has the same label as vc, whch s the same as that of ab. The other cases are smlarly verfed. Ths procedure s called the consstency check. We have thus shown: Theorem 8.2. The total tme and space complexty for the consstency check s lnear. 9. Factorzng by mergng colors Now we have prepared all the tools we need for the prme factorzaton of a gven connected graph G. Bythe refnement Lemma t suffces to compute the fnest product relaton of the gven graph G. The colorng t nduces s the product colorng of the unque prme factor decomposton. It s the fnest edge colorng satsfyng the Isomorphsm Property.

11 482 W. Imrch, I. Petern / Dscrete Mathematcs 307 (2007) Theorem 9.1. The prme factorzaton of connected graphs can be found n lnear tme and space. Proof. We have to present an algorthm. The dea of the algorthm s to start wth a colorng that s not coarser than the fnest product colorng and to merge color classes when necessary. Snce we can color a product f we know the colors of the edges ncdent wth a gven vertex v 0 we start wth the relaton that assgns dfferent colors to the edges ncdent wth v 0. Snce every vertex meets edges of all colors no colors can be mssed. It s practcal to choose v 0 among the vertces of mnmal degree, there cannot be more than d(v 0 ) color. We call these colors ntal colors henceforth. We choose v 0 as the bass of the BFS orderng and run the labellng algorthm and the consstency check. It s possble that both go through. Then the result s the colorng wth respect to the prme factor decomposton of G. Most lkely though the labellng algorthm wll run nto dffcultes or the consstency check. For example, f there are cross-edges n L 1 the labellng algorthm breaks down. So, what should we do? The answer s easy, every cross edge n L 1 has to have the same color as ts down-edges. Snce ths s not so, our colorng s too fne, we have too many colors and merge them whenever necessary. It wll be useful not to recolor any edges though, we smply note the ntal colors that have been merged. Snce d(v 0 ) 2 d(v 0 )n 2m we need no sophstcated data structure to do ths from the pont of vew of space requrements. Also, snce we cannot have less than one color, we have at most d(v 0 ) merge operatons n whch two sets (of ntal colors) of total sze less than d(v 0 ) are merged. If we order the sets, then ths can be done n d(v 0 ) steps. So the total tme needed s lnear n m too. The ntal color of smallest ndex n a set of merged colors wll be called ts prncpal color, t s the representatve of the set of merged colors. These sets are the new colors. An array of colors as descrbed n the labellng algorthm wll thus consst of subarrays of ntal colors and the labels wll be gven wth respect to the ntal colors, of course always keepng n mnd to whch (prncpal) color they belong. In general we wll have checked the consstency of the colorng up to level L and wsh to contnue. The frst step s the labellng algorthm. When t fals we have to merge colors. A short analyss of the reasons why t may fal shows that there may be mssng vertces, mssng edges, too many vertces or too may edges or an array nto whch the algorthm tres to put an edge s too short. These cases have been analyzed n [6, p. 237], the answer s always the same: f somethng goes wrong the vertex just beng consdered must be classfed as a unt-layer vertex to allow the labellng algorthm to contnue. Ths means that all ntal colors of the down- and cross-edges of ths vertex have to be merged. It s clear that we do not have to recolor any edges. Even more mportantly, we do not have to rerun the consstency check. The reason s that fewer colors mean fewer and larger layers. Hence, fewer consstency checks are necessary and all the ones performed retan ther valdty. After the labellng algorthm has been completed for L +1 we run the consstency check. Agan, f somethng goes wrong, the reasons can only be the same as before and we merge colors. The result s, that the consstency checks that dd not work just do not have to be performed any more! We conclude the paper wth a short summary of the man steps of our algorthm. Algorthm Intalzaton Choose a vertex v 0 of mnmum degree and execute the BFS algorthm wth base v Label the up-edges of v 0 (wth name 1 and dstnct colors). 2. For every BFS level L, = 1, 2,...,r 1 do 2.1. Label (down-, cross-, and up-) edges of L Merge colors f necessary Check for the consstency Merge colors f necessary. Acknowledgment The senor author wshes to express hs grattude to L. Baba who asked hm n 1982 whether there exsted a polynomal algorthm for the prme factorzaton of connected graphs. Thanks are also due to Ross McConnell and Danel Varga for nterestng dscussons.

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