Fast Skew Partition Recognition

Size: px

Start display at page:

Download "Fast Skew Partition Recognition"

Lynette Hood
6 years ago
Views:

1 Fast Skew Partition Recognition William S. Kennedy 1, and Bruce Reed 2, 1 Department of Mathematics and Statistics, McGill University, Montréal, Canada, H3A2K6 kennedy@math.mcgill.ca 2 School of Computer Science, McGill University, Montréal, Canada, H3A2A7 breed@cs.mcgill.ca Abstract. Chvátal defined a skew partition of a graph G to be a partition of its vertex set into two non-empty parts A and B such that A induces a disconnected subgraph of G and B induces a disconnected subgraph of G. Skew partitions are important in the characterization of perfect graphs. De Figuereido et al. presented a polynomial time algorithm which given a graph either finds a skew partition or determines that no such partition exists. It runs in O(n 101 ) time. We present an algorithm for the same problem which runs in O(n 4 m)time. 1 Introduction A skew partition of a graph G =(V,E) is a partition of V into two nonempty sets A and B such that G[A] is not connected and G[B] is not connected. If such a partition exists, B is called a skew cutset. Clearly, a skew partition (A, B) of G yields a skew partition (B,A) ofg. Itisthisself-complementaritywhich first suggested that these partitions might be important to an understanding of the structure of perfect graphs. A graph is perfect if each induced subgraph has chromatic number equal to the size of its largest clique. A graph is Berge if it contains neither an induced odd chordless cycles of length at least five or the complement of such a cycle. Berge introduced these two classes of graph and proposed the Strong Perfect Graph Conjecture that, in fact, they are identical [1]. Speculating that skew partitions might play a key role in a decomposition theorem for Berge graphs which would imply the Strong Perfect Graph Conjecture, Chvátal [3] introduced skew partitions and conjectured that no minimal imperfect graph permits a skew partition. Both his speculation and his conjecture were accurate. Indeed, Chudnovsky, Robertson, Seymour, and Thomas [2] recently proved every Berge graph either: (a) Is in one of five basic classes of perfect graphs (line graphs of bipartite graphs, their complements, bipartite graphs, their complements, or double split graphs), or Research supported by NSERC doctoral fellowship. Research supported in part by NSERC Canada Research Chair. H. Ito et al. (Eds.): KyotoCGGT 2007, LNCS 4535, pp , c Springer-Verlag Berlin Heidelberg 2008

2 102 W.S. Kennedy and B. Reed (b) Permits one of three partitions (a proper 2-join, a homogeneous pair, or a special type of skew partition which they call balanced 1. It was known that the first two of these three partitions could not occur in a minimal imperfect graph (see [5] and [4]). Chudnovsky et al. also proved that balanced skew partitions cannot occur in a smallest minimal imperfect Berge graph. These results taken together imply the Strong Perfect Graph Conjecture. This paper presents a polynomial time algorithm to test if a graph has a skew partition. This is not the first such algorithm. In [6], de Figuereido, Klein, Kohayakawa and Reed present one whose running time is O(n 101 ). In this paper we present an algorithm with running time O(n 4 m). We note that in related work, Trotignon has concurrently and independently developed an O(n 9 ) algorithm for determining if a Berge graph has a balanced skew cutset and proves that determining if an arbitrary graph has such a cutset is NP-hard [11]. In 2, we present a preliminary discussion of our method for finding skew partitions. In 3, we present our O(n 4 m) time algorithm to find a skew partition. We assume the reader is familar with the standard definitions and notations of perfect graph theory which can be found in [8]. We warn the reader that, following the conventions of that field, by a subgraph we mean an induced subgraph. 2 The Idea Our algorithm breaks the problem up into two subproblems. We first check if the graph has a special kind of skew partition known as a T -cutset. We then look for skew partitions which are not T -cutsets. T -cutsets are easy to handle, so we treat them first. Hoàng [7] defined a T -cutset as a skew cutset B where there exist two vertices x and y, such that x and y are in different components of G B and some component of G[B] is contained in N(x) N(y). For a subset of the vertex set S = {s 1,s 2,...,s i },letn(s) = i j=1 N(s j). Lemma 1. There is an O(n 3 m) time algorithm to decide whether a graph G contains a T -cutset. Proof. Consider every pair of vertices x, y and component C of G[N(x) N(y)]. If B is a T -cutset separating x from y such that C is a component of G[B] then B C N(C) x y. Moreover,G[C N(C) x y] is disconnected. So C N(C) x y is also a T -cutset separating x from y. Givenx, y and C we can test if C N(C) x y is a T -cutset in O(m) time (by testing the connectivity of G[V (C N(C) x y)]). So, we can test if any T -cutset corresponding to such a triple exists in O(n 3 m). 1 A skew partition (A, B) isbalanced if every path P = {v 1,v 2,..., v k 1,v k } in G of length at least 2 such that v 1 and v 2 are in B and v 2,..., v k 1 are in A has even length, and if every path P = {v 1,v 2,..., v k 1,v k } in G of length at least 2 such that v 1 and v 2 are in A and v 2,..., v k 1 are in B has even length.

3 Fast Skew Partition Recognition 103 Our approach to looking for skew cutsets which are not T -cutsets is motivated by Reed s algorithm [9] for finding skew cutsets in bipartite graphs (in which case they are complete bipartite subgraphs). So, we first briefly sketch the ideas of this algorithm. Suppose we have a bipartite graph with bipartition (S, T ). Reed s approach was to ask, for each k, if there is a skew cutset B with B T = k. Sincewe know G has no T -cutsets, we can restrict our attention to skew cutsets B with N(x) N(y) <kfor every pair of vertices x and y of S in different components of G B. On the other hand, if x and y are two vertices of B S then the intersection of their neighbourhoods contains B T and thus has at least k vertices. Hence, B S is a clique cutset in the auxiliary graph whose vertex set is S and where two vertices are adjacent if their neighbourhoods intersect in at least k vertices. Therefore, we need only check if this auxiliary graph contains a clique cutset which corresponds to a skew cutset of G. As not every clique cutset of the auxiliary graph corresponds to a skew cutset, we use an algorithm due to Tarjan [10] to construct a structure known as a clique cutset tree, and then do a bit more work, to determine if the auxiliary graph has the desired special clique cutset. We omit further details in this preliminary discussion. Bipartite graphs are rather special in that any skew cutset B consists of exactly two stable sets. Furthermore, every vertex sees vertices in only one stable set of B. So, bounding the size of the stable sets of B bounds the size of N(x) N(y) forx and y in different components of G C. This is not true in general graphs which complicates our cutset finding algorithm. For each vertex r of the graph and every pair of integers k 1 and k 2,with k 2 k 1 we ask if there is a skew cutset B such that (a) r B, (b) some largest component L of G[B] hassizek 1, r L, and (c) the component U of G[B] containing r has size k 2. Given such a skew cutset B, for any two nonadjacent vertices x and y of B, either N(x) N(y) containsv (L) and hence (i) some component of G[N(x) N(y)] has size at least k 1, or x and y are in L and hence V (U) is contained in N(x) N(y), so (ii) some component of G[N(x) N(y)] contains r and has size at least k 2. On the other hand, if x and y are in different components of G B then they are not adjacent and N(x) N(y) B. The key lemma is the following: Lemma 2. Suppose B is a skew cutset of G which is not a T -cutset and B satisfies (a) (c). If x, y are vertices of different components of V B, then neither (i) nor (ii) holds for x, y. As in the bipartite case, we let H be an auxiliary graph with vertex set V where vertices x and y are adjacent if they are adjacent in G or if they satisfy either

4 104 W.S. Kennedy and B. Reed (i) or (ii). Lemma 2 implies that such a skew cutset B is a clique cutset of H. As we describe in the next section, we use these ideas to develop an algorithm for finding the desired skew cutset. 3 The Details This section presents the details of the skew partition algorithm for general graphs sketched in the previous section. In particular, we prove our main result: Theorem 1. Let G be a graph. There exists an O(n 4 m) algorithm to find a skew cutset in G or decide that no such cutset exists. By Lemma 1, we can check if G contains a T -cutset in O(n 3 m) time. Henceforth, we assume G contains no T -cutset. For each vertex r and each pair of integers 0 k 2 k 1 n k 2,wesearchforaskewcutsetB satisfying (a) (c). In Lemma 4, we show for each such triple (k 1,k 2,r)anO(nm) algorithm to check if the desired skew cutset exists or return that no such cutset exists. Thus, our algorithm to find a skew cutset in G or decide that no such cutset exists takes O(n 4 m)time. We create an auxiliary graph H whose vertex set is V (G) and for which xy is an edge of H if either xy is an edge of G or one of (i) and (ii) holds for x, y. The previous section showed B induces a clique cutset in H. As not every clique cutset of H is a skew cutset of G, we need to check if H contains a clique cutset whose vertices induce a skew cutset of G. To do so, we use an auxiliary structure known as a clique cutset tree. A clique cutset tree for a graph F consists of a rooted tree T with root r such that (I) every node t of T is labelled with a subgraph F t of F,inparticular, F r = F,(II)ifnodet has children s 1,...,s i,thenf t = F s1... F si and F s1... F si, which we denote K t, is a clique cutset of F t, and (III) if l is a leaf of T then F l has no clique cutset. We note that clique cutset trees have been well studied in the context of perfect graphs and refer the interested reader to [8] for more details. Let T be a clique cutset tree for H. Fortherootw of T, trivially, H w contains every clique cutset of H. For any leaf l of T, by definition, H l does not contain any clique cutset. On the other hand, every clique is contained in some leaf of the clique cutset tree. Thus, for every clique cutset C there exists a node s of T such that C is contained in H s, C is not a cutset of H s and C is a cutset of H t for the parent t of s. Thus, our clique cutset K, if it exists, corresponds to a skew cutset of G for which conditions (a) (c) from 2 hold for the triple (r, k 1,k 2 ), and such that for some node s of T (d) K is contained in H s, (e) K is not a cutset of H s,and (f) K is a cutset of H t,wheret is the parent of s. In looking for skew cutsets satisfying (d) (f) for a node s with parent t in T, the following will be useful:

5 Fast Skew Partition Recognition 105 Lemma 3. Let K be a clique cutset in H whose vertices induce a skew cutset B in G such that conditions (a) (f) are satisfied for some node s of T with parent t. Thens has a sibling u in T such that H s K and H u K t are in different components of G B. Proof. Let R = H s K.By(e),R is connected in H. We now show R is contained in the same component of G B. Assume this is not the case and let C 1 and C 2 be two components of G B such that C 1 R and C 2 R. Letc 1 be in C 1 R and c 2 be in C 2 R. AsR is connected, there exists some path P H = p 1 p 2...p k in R where p 1 = c 1 and p k = c 2.Letp i be the first vertex not contained in C 1.But,asp i 1 p i E(G) andp i 1 p i E(H), p i 1 and p i must therefore satisfy (i) or (ii) and contradict Lemma 2. This contradiction shows that R is indeed in a single component of G B. Let the other children of t be u 1,...,u k,andfori =1,...,klet R i = H ui K t. Every component R i is disjoint from H s and, by (d), disjoint from K. Byan identical argument presented for R, R i is contained in a single component of G B. Finally, as K is a clique cutset of H t and R is connected in G B there exists some i for which R i and R are in different components of H t K. As E(G) E(H) andk corresponds to B, it follows that R i and R are in different components of G B. This claim leads to the following algorithm. Lemma 4. There is an O(nm) time algorithm to decide whether there exists a clique cutset K of H corresponding to a skew cutset in G satisfying (a) (c). Proof. We begin by building a clique cutset tree T for H, whichasshownby Tarjan [10], can be constructed in O(nm) time. Tarjan s algorithm constructs a clique cutset tree with at most n 1 leaf nodes and such that every internal node of this tree has exactly two children. For each node s where H s is a clique, we first check if H s induces a skew cutset in G in O(n + m) time and henceforth assume that if K exists it is properly contained in H s for some s. WethenuseanO(n + m) time algorithm which decides for a fixed sibling pair s, u in T with parent t if there exists a clique cutset K corresponding to a skew cutset in G satisfying (a) (f) such that H s K and H u K t are in different components of G B. Either we find the desired K or we check all sibling pairs, in which case Lemma 3 implies G contains no skew cutset. As there are at most n 1 sibling pairs, we have the desired O(nm) time algorithm. All that remains to be shown is the O(n + m) algorithm. For fixed siblings s, u in T,lett be their parent. Our algorithm maintains two vertex sets: the set R containing vertices which must be in the same component of G B as H u K and the set K containing vertices of H s which must be in K. Initially let R and K be empty. We use the standard depth first search algorithm with the following changes: 1) we start the search from any unseen node in H u K t, 2) for each node seen which is not contained in H s we add it to R and continue the search as normal from this node, and 3) for each node seen in H s we add it to K and do not continue searching any deeper from this

6 106 W.S. Kennedy and B. Reed node. If our search ends, then if H u (K t R ) we restart the search at any node in H u (K t R ), otherwise we terminate the search. These modification maintain the O(n + m) running time of depth first search. After the search if K = H s then the desired skew cutset does not exist. If not then the set K is a cutset and we need only check if it is contained in a skew cutset in G. IfG[K ] is disconnected then K is itself a skew cutset. Otherwise, if K is part of a skew cutset then K must be contained in one of its components. Let J = N(K ) H s.ifj =, then the desired skew cutset does not exist. If (K J) H s,then(k J) is the desired skew cutset. If (K J) =H s and J = 1 then the desired skew cutset does not exist. Otherwise, if J > 1then for any vertex v of J, wehavethatk {v} is the desired skew cutset. It follows these steps take O(n + m) running time, so together with the search we have the desired O(n + m) algorithm. 4 Concluding Remarks Reed s algorithm for finding a clique cutset in a bipartite graph [9] does not use Tarjan s algorithm [10] for finding a clique cutset. We can improve the run time of this algorithm to O(n 2 m) by using the ideas presented in Section 2 and Lemma 4. This follows from the proof of Reed s algorithm as its run time is O(n (f 1 + f 2 )) where f 1 is the run time for constructing a clique cutset tree and f 2 is run time for the algorithm presented in Lemma 4. Acknowledgement The authors would like to thank Sulamita Klein and Chinh Hoang for their helpful comments. References 1. Berge, C.: Les problèmes de coloration en théorie des graphes. Publ. Inst. Stat. Univ. Paris 9, (1960) 2. Chudnovsky, M., Robertson, N., Seymour, P., Thomas, R.: The strong perfect graph theorem. Annals of Mathematics 164, (2006) 3. Chvátal, V.: Star-cutsets and perfect graphs. J. Combin. Theory Ser. B 39, (1985) 4. Chvátal, V., Sbihi, N.: Bull-free berge graphs are perfect. Graphs and Combinatorics 3, (1987) 5. Cornuéjols, G., Cunningham, W.: Compositions for perfect graphs. Discrete Math. 55, (1985) 6. de Figuereido, C.M.H., Klein, S., Kohayakawa, Y., Reed, B.A.: Finding skew partitions efficiently. J. Algorithms 37, (2000) 7. Hoàng, C.: Some properties of minimal imperfect graphs. Discrete Math. 160, (1996) 8. Ramirez-Alfonsin, J., Reed, B.A. (eds.): Perfect graphs. J.H. Wiley, Chichester (2001)

7 Fast Skew Partition Recognition Reed, B.A.: Skew partitions in perfect graphs. In: Discrete Applied Mathematics (2005) (in press) 10. Tarjan, R.E.: Decomposition by clique separators. Discrete Math. 55, (1985) 11. Trotignon, N.: Decomposing berge graphs and detecting balanced skew partitions. Journal of Combinatorial Theory Series B 98, (2008)

Skew partitions in perfect graphs

Discrete Applied Mathematics 156 (2008) 1150 1156 www.elsevier.com/locate/dam Skew partitions in perfect graphs Bruce Reed a,b a McGill University, Montreal, Canada b CNRS, France Received 14 June 2005;