On the complexity of finding a sun in a graph

arXiv:0807.0462v1 [cs.DM] 3 Jul 2008 On the complexity of finding a sun in a graph Ch´ınh T. Ho`ang∗ Abstract The sun is the graph obtained from a cycle of length even and at least six by adding edges to make the even-indexed vertices pairwise adjacent. Suns play an important role in the study of strongly chordal graphs. A graph is chordal if it does not contain an induced cycle of length at least four. A graph is strongly chordal if it is chordal and every even cycle has a chord joining vertices whose distance on the cycle is odd. Farber proved that a graph is strongly chordal if and only if it is chordal and contains no induced suns. There are well known polynomial-time algorithms for recognizing a sun in a chordal graph. Recently, polynomial-time algorithms for finding a sun for a larger class of graphs, the so-called HHD-free graphs, have been discovered. In this paper, we prove the problem of deciding whether an arbitrary graph contains a sun in NP-complete. Keywords: chordal graph, strongly chordal graph, sun 1 Introduction A hole is an induced cycle with at least four vertices. A graph is chordal if it does not contain a hole as an induced subgraph. Farber [6] defined a graph to be strongly chordal if it is chordal and every cycle in the graph on 2k vertices, k ≥3, has a chord uv such that each segment of the cycle from u to v has an odd number of edges. We denote by k-sun the graph obtained from a cycle of length 2k (k ≥3) by adding edges to make the even-indexed vertices pairwise adjacent. Figure 1 shows a 5-sun. A sun is simply a k-sun for some k ≥3. Farber showed [6] that a graph is strongly chordal if and only if it is chordal and does not contain a sun as induced subgraph. Farber’s motivation was a polynomial-time algorithm for the minimum weighted dominating set problem for strongly chordal graphs. The problem is NP-hard for chordal graphs [1]. In this paper, we prove that it is NP-hard to find a sun in an arbitrary graph. This result is motivated by the following discussion on chordal and strongly chordal graphs. For more information on this topics, see [3, 7]. We use N(x) to denote the set of vertices adjacent to vertex x in a graph G. Define N[x] = N(x) ∪{x}. A vertex x in a graph is simplicial if N(x) induces a complete graph. It is well known [4] that graph G is chordal if and only if every induced subgraph H of G contains a simplicial vertex of H. Farber proved [6] an analogous characterization for strongly chordal graphs. A vertex x in a ∗Department of Physics and Computer Science, Wilfrid Laurier University, Canada. Research supported by NSERC. email: choang@wlu.ca 1 Figure 1: The 5-sun HOUSE HOLE DOMINO Figure 2: The house, the hole and the domino graph is simple if the vertices in N(x) can be ordered as x1, x2, . . . , xk such that N[x1] ⊆N[x2] ⊆ . . . ⊆N[xk]. Thus, every simple vertex is simplicial. For a graph G, let R = v1, v2, . . . , vn be an ordering of vertices of G. Let G(i) = G[{vi, vi+1, . . . , vn}], i.e., the subgraph induced in G by the set vi through vn of vertices. R is a simple elimination ordering for G if vi is simple in G(i), 1 ≤i ≤n. The following is due to Farber [6]: Theorem 1 ([6]) The following are equivalent for any graph G: • G is strongly chordal. • G is chordal and does not contain a sun. • Vertices of G admit a simple elimination ordering. Thus, suns play an important role in the studies of chordal and strongly chordal graphs. There are well known algorithms [17, 12] to test whether a chordal graph is strongly chordal and thus whether it contains a sun. It is natural to investigate the problem for larger classes of graphs. A graph is HHD-free if it does not contain a house, a hole, or a domino (see figure 2). Every chordal graph is a HHD-free graph. HHD-free graphs [10] have several properties analogous to those of chordal graphs. Brandst¨adt [2] proposed the problem of finding a sun in a HHD-free graph. This problem was proved to be polynomial-time solvable in [13] and [5]. In this paper, we will prove the following Theorem 2 It is NP-complete to decide whether a graph contains a sun. 2 Denote by k-hole the hole on k vertices. A k-antihole is the complement of a k-hole. A graph is weakly chordal [8] if it does not contain a k-hole or k-antihole with k ≥5. Weakly chordal graphs generalize chordal graphs in a natural way, and they are known to be perfect and have many interest- ing algorithmic properties (see [9]). In spite of Theorem 2, it is conceivable there are polynomial- time algorithms to solve the sun recognition problem for weakly chordal graphs or even perfect graphs [15]. In this spirit, we will refine Theorem 2 to obtain a stronger result. Theorem 3 It is NP-complete to decide whether a graph G contains a sun, even when G does not contain a k-antihole with k ≥7. Let k-CLIQUE (respectively, k-SUN) be the problem whose instance is a graph G and an integer k, for which the question to be answered is whether G contains a clique on k vertices (respectivel

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