View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by RERO DOC Digital Library Journal of Combinatorial Optimization, 10, 211–225, 2005 c 2005 Springer Science + Business Media, Inc. Manufactured in The Netherlands. On Split-Coloring Problems T. EKIM tinaz.ekim@epfl.ch D. de WERRA dewerra.ima@epfl.ch Institute of Mathematics—ROSE, Ecole Polytechnique Fed´ erale´ de Lausanne, CH-1015 Lausanne-Ecublens, Switzerland Received November 19, 2003; Revised June 1, 2005; Accepted June 2, 2005 Abstract. We study a new coloring concept which generalizes the classical vertex coloring problem in a graph by extending the notion of stable sets to split graphs. First of all, we propose the packing problem of finding the split graph of maximum size where a split graph is a graph G = (V, E)inwhich the vertex set V can be partitioned into a clique K and a stable set S.Nocondition is imposed on the edges linking vertices in S to the vertices in K . This maximum split graph problem gives rise to an associated partitioning problem that we call the split-coloring problem. Given a graph, the objective is to cover all his vertices by a least number of split graphs. Definitions related to this new problem are introduced. We mention some polynomially solvable cases and describe open questions on this area. Keywords: split-coloring, vertex covering by split graphs, partitioning, packing 1. Introduction Packing and partitioning problems in graphs have been widely studied by many authors for years. Maximum stable set and maximum clique problems are the basic packing problems and their associated partitioning problems are called respectively graph coloring and vertex covering by cliques. These problems, especially the graph coloring have many applications in scheduling (de Werra et al., 2005), timetabling, etc. but on the other hand, they are known to be among the most difficult NP-hard problems. In this paper, we define a new problem which includes the graph coloring problem and extends its field of application. First of all, we consider the problem of finding in a graph G an induced split graph of maximum size. A split graph is a graph G = (V, E)inwhich the vertex set V can be partitioned into a clique K and a stable set S.Nocondition is imposed on the edges linking vertices in S to the vertices in K . This maximum split graph problem gives rise to an associated partitioning problem that we call the split-coloring problem. It is a vertex coloring problem of a graph where we replace stable sets by split graphs. Given a graph, the objective is to cover all vertices by the least number of induced split graphs. This problem is motivated both by applications and by theoretical properties as auto- complementarity and heredity. As a related application of the concept of split graph, we can mention a telecommunication problem which consists of assigning terminal nodes to concentrators and install concentrators and links in order to ensure an optimal traffic routing. We observe that split graphs appear in this problem since concentrators are represented by 212 EKIM AND de WERRA cliques and terminal nodes assigned to a same concentrator constitute a stable set (Gourdin et al., 2002). The paper is organized as follows : in the second section some preliminary results are presented. We give an integer programming model of the split-coloring problem in section three. The fourth section is the resolution of the split-coloring problem in some special classes of graphs as for instance cacti. In section five, we give lower and upper bounds for χS in arbitrary graphs. Finally, the last section mentions some open questions for further research. In what follows, Kl denotes a clique on l vertices, Cl acycle on l vertices and Pl a path on l vertices. 2. Preliminary results Split graphs are characterized as follows: Theorem 1 (F¨oldes and Hammer, 1976). Forevery graph G = (V, E), the following conditions are equivalent: 1. Gisasplit graph, 2. G and Ga¯ re triangulated, 3. G does not contain 2K2, C4 or C5. One can remark that as the complement of a split graph is again a split graph, solving the split-coloring problem in a graph is equivalent to solve it in the complementary graph. More generally, if we can solve it in a class of graphs C then we can also solve it in the class C containing the complements of all graphs in C and vice versa, thereby the problem of split-coloring the vertices of a graph is an auto-complementary problem. On the other hand, we know that if G is a split graph then all subgraphs of G are also split graphs. This hereditary property with respect to vertices can be effectively exploited in the conception of algorithms. The split-coloring problem is a very intriguing one because it generalizes the classical graph coloring problem and furthermore, it has never been studied systematically to our knowledge. A corresponding edge covering problem has been studied though (see Mahadev and Peled, 1995). Let us call a subset S of vertices split-independent (s.i.) if the subgraph induced by S is a split graph. Then, our basic packing problem called maximum split graph, consists in finding the split-independence number αS(G)ofG, which is the maximum cardinality |S| of a s.i. set S.Inother words, we search for a maximum size induced split graph in G.It should be trivially noted that αS(G) ≥ max(α(G),ω(G)) + 1 where α(G)isthe stability number and ω(G)isthe maximum clique size. Moreover, one can observe that for any graph G,wehaveα(G) + ω(G) − 1 ≤ αS(G) ≤ α(G) + ω(G) since a maximum clique and a maximum stable set have at most one vertex in common. The problem of finding αS(G)is obviously NP-hard: Theorem 2. Forafixed k and a given graph G, it is NP-complete to determine whether αS(G) ≥ k. ON SPLIT-COLORING PROBLEMS 213 Proof: The maximum clique problem is reduced to maximum split graph problem. Let the graph G = (V, E) (which is not a stable set) be an instance of maximum clique problem. We will consider the graph G = G ⊕ IN obtained from G by adding a stable set IN of size N ≥|V | which is completely linked to G, i.e., any vertex of V is linked to every vertex of IN . Assume there is an algorithm for the maximum split graph problem which gives αS(G ) when applied to G .Now, note that IN is the only maximum stable set in G .Onthe other hand, any maximum clique of G is in the form Kmax ∪{x} where Kmax is a maximum clique of G and where x ∈ IN . This implies that there are no disjoint maximum clique and maximum stable set in G , hence the graph induced by Kmax ∪ IN is a maximum split graph in G of cardinality αS(G ) = ω(G ) + N − 1 = ω(G) + N. The size of a maximum clique in G could be then easily computed by the formula ω(G) = αS(G ) − N. But this is not possible since maximum clique is NP-complete, consequently, so it is the maximum split graph problem. The split-coloring problem consists in minimizing the number of split graphs which cover all the vertices of G. This minimum number is called the split-chromatic number and we denote it by χS.Atrivial upper bound for χS(G)ismin(χ(G),θ(G)) where χ(G)is the chromatic number and θ(G)isthe minimum number of cliques covering the vertices of G.Wesay that a graph G = (V, E)isk-split-colorable if there exists a partition of V into k split graphs. Although split graphs had been extensively studied by many authors (see F¨oldes and Hammer, 1976, 1977; Hammer and Simeone, 1981; Benzaken et al., 1985; Chernyak and Chernyak, 1991), this problem does not seem to have been studied to our knowledge. The only coloring problem related to split graphs which appears extensively in the literature deals with the case where the edge set of a graph has to be covered (Chernyak and Chernyak, 1991; Mahadev and Peled, 1995). However, this problem of edge coloring by split graphs has neither the hereditary character nor the auto-complementarity property mentioned above. We know that split graphs can be recognized in O(|V |2) time (Mahadev and Peled, 1995). In other words, the problem of determining for a given graph G = (V, E) 2 whether χS(G) = 1 can be solved in O(|V | ) time. Now, let us define a new notion which will be useful in our study. Definition 1. Agraph G is called k-split-critical if and only if χS(G) = k and ∀v, χS(G − v) = k − 1. According to this definition, 2K2, C4 and C5 are obviously the unique 2-split-critical structures. Let H be a graph, we denote by kH a graph having k connected components isomorphic to H.Inwhat follows H will be replaced by an odd cycle or a clique. By abuse of language, we often call kH, the graph consisting of k induced H.Inorder to characterize some 3-split- critical graphs and as a preliminary for further developments, we mention the following three facts. In what follows, OC denotes an odd cycle, i.e., a cycle on an odd number of vertices. Fact 1. For any odd cycle OC of length at least 5, we have χS(OC) = 2; a 2-split-coloring is obtained by choosing one clique (of size one or two) and two stable sets.