ON THE THRESHOLD-WIDTH OF GRAPHS Maw-Shang Chang1, Ling-Ju Hung1, Ton Kloks⋆, and Sheng-Lung Peng2 1 Department of Computer Science and Information Engineering National Chung Cheng University Min-Hsiung, Chia-Yi 621, Taiwan (mschang,hunglc)@cs.ccu.edu.tw 2 Department of Computer Science and Information Engineering National Dong Hwa University Shoufeng, Hualien 97401, Taiwan [email protected] Abstract. The G-width of a class of graphs G is defined as follows. A graph G has G-width k if there are k independent sets N1,..., Nk in G such that G can be embedded into a graph H ∈ G such that for every edge e in H which is not an edge in G, there exists an i such that both endpoints of e are in Ni. For the class TH of threshold graphs we show that TH-width is NP-complete and we present fixed-parameter algorithms. We also show that for each k, graphs of TH-width at most k are characterized by a finite collection of forbidden induced subgraphs. 1 Introduction Definition 1. Let G be a class of graphs which contains all cliques. The G-width of a graph G is the minimum number k of independent sets N1,..., Nk in G such that there exists an embedding H ∈ G of G such that for every edge e = (x, y) in H which is not an edge of G there exists an i with x, y ∈ Ni. We restrict the G-width parameter to classes of graphs that contain all cliques to ensure that it is well-defined for every (finite) graph. In this paper we investigate the width-parameter for the class TH of threshold graphs and henceforth we call it the threshold-width or TH-width. If a graph G has threshold-width k then we call G also a k-probe threshold graph. We refer to the partitioned case of the problem when the collection of independent sets arXiv:1210.8365v1 [math.CO] 30 Oct 2012 Ni, i = 1, . , k, which are not necessarily disjoint, is a part of the input. A collection of independent sets Ni, i = 1, . , k, is a witness for a partitioned P k N graph. For historical reasons we call the set of vertices = V − Si=1 i the set k N of probes and the vertices of Si=1 i the set of nonprobes. Threshold graphs were introduced in [3] using a concept called ‘threshold di- mension.’ There is a lot of information about threshold graphs in the book [15], and there are chapters on threshold graphs in the book [6] and in the survey [1]. We may take the following characterization as a definition [3, 1]. ⋆ This author is supported by the National Science Council of Taiwan, under grants NSC 98–2218–E–194–004 and NSC 98–2811–E–194–006. Definition 2. A graph G is a threshold graph if G and its complement G¯ are trivially perfect. Equivalently, G is a threshold graphs if G has no induced P4, C4, nor 2K2. We end this section with some notational conventions. For two sets A and B we write A + B and A − B instead of A ∪ B and A \ B. We write A ⊆ B if A is a subset of B with possible equality and we write A ⊂ B if A is a subset of B and A 6= B. For a set A and an element x we write A + x instead of A + {x} and A − x instead of A − {x}. In those cases we will make it clear in the context that x is an element and not a set. A graph G is a pair G = (V, E) where the elements of V are called the vertices of G and where E is a set of two-element subsets of V, called the edges. We denote edges of a graph as (x, y) and we call x and y the endvertices of the edge. For a vertex x we write N(x) for its set of neighbors and for W ⊆ V we write N(W)= Sx∈W N(x)−W for the neighbors of W. We write N[x]= N(x)+x for the closed neighborhood of x. For a subset W we write N[W]= N(W)+ W. Usually we will use n = |V| to denote the number of vertices of G and we will use m = |E| to denote the number of edges of G. For a graph G = (V, E) and a subset S ⊆ V of vertices we write G[S] for the subgraph induced by S, that is the graph with S as its set of vertices and with those edges of E that have both endvertices in S. For a subset W ⊆ V we will write G−W for the graph G[V −W] and for a vertex x we will write G−x rather than G − {x}. We will usually denote graph classes by calligraphic capitals. In the next section we show that the class of graphs with TH-width at most k is characterized by a finite collection of forbidden induced subgraphs. 2 A finite characterization A graph is a threshold graph if and only if it has no induced P4, C4,or2K2. We show that for any k, the class of graphs with TH-width at most k is characterized by a finite collection of forbidden induced subgraphs. Lemma 1. A graph G is a threshold graph if and only if it has a binary tree- decomposition (T, f), where f is a bijection from the vertices of G to the leaves of T. Every internal node of T, including the root is labeled either as a join– or a union- node. For every internal node the right subtree consists of a single leaf. Two vertices are adjacent in G if their lowest common ancestor in T is a join-node. Proof. According to Theorem 2 on the facing page a graph is a threshold graph if and only if every induced subgraph has either a isolated vertex or a universal vertex. This proves the lemma. ⊓⊔ Theorem 1. For every k the class of graphs with TH-width at most k is character- ized by a finite collection of forbidden induced subgraphs. Proof. To prove this theorem we use the technique introduced by Pouzet [23]. 2 Obviously, the class of graphs with TH-width at most k is hereditary. Let k be fixed. Assume that the class of TH-width at most k has an infinite collection of minimal forbidden induced subgraphs, say G1, G2,.... In each Gi single out ′ ′ one vertex ri and let Gi = Gi − ri. Then Gi has TH-width at most k, thus there N(i) N(i) ′ ′ are independent sets 1 ,..., k in Gi such that Gi can be embedded into a threshold graph Hi by adding certain edges between vertices that are pairwise N(i) contained in some ℓ . For each i consider a binary tree-decomposition (Ti, fi) for Hi as stipulated in Lemma 1. Each leaf is labeled by a 0/1-vector with k entries. The jth entry of this vector is equal to 0 or 1 according to whether the N(i) ′ vertex is contained in j or not. Thus two vertices are adjacent in Gi if and only if their lowest common ancestor is a join-node and their vectors are disjoint. We give each leaf an additional 0/1-label that indicates whether the vertex that is mapped to that leaf is adjacent to ri or not. Kruskal’s theorem [14] states that binary trees, with points labeled by a well- quasi-ordering are well-quasi-ordered with respect to their lowest common an- cestor embedding. When we apply Kruskal’s theorem to the labeled binary trees ′ Ti that represent the graphs Gi we may conclude that there exist i<j such that ′ ′ Gi is an induced subgraph of Gj. But then we must also have that Gi is an in- duced subgraph of Gj. This is a contradiction because we assume that the graphs Gi are minimal forbidden induced subgraphs. This proves the theorem. ⊓⊔ 3 TH-width is fixed-parameter tractable In this section we show that for constant k, k-probe threshold graphs can be recognized in O(n3) time. The following is one of the many characterizations of threshold graphs. Theorem 2 ([4,8,16,19]). A graph is a threshold graph if and only if every induced subgraph has an isolated vertex or a universal vertex.3 The following theorem is a monadic second-order characterization. Problems that can be formulated in monadic second-order logic can be solved on graphs that have bounded rankwidth [5]. We will show that k-probe threshold graphs have bounded rankwidth shortly. Theorem 3. A graph G = (V, E) has threshold-width at most k if and only if there exist k independent sets Ni, i = 1, . , k, such that for every W ⊆ V, G[W] has an isolated vertex or a vertex ω such that for every y ∈ W − ω either ω is adjacent to y or there exists i ∈ {1, . , k} with {ω, y} ⊆ Ni. Proof. This is inferred by Theorem 2 and Definition 1. ⊓⊔ 3 A vertex x of a graph G is isolated if its neighborhood is the empty set. A vertex x of a graph G =(V, E) is universal if N[x]= V. 3 Definition 3 ([22, 20]). A rank-decomposition of a graph G = (V, E) is a pair (T, τ) where T is a ternary tree and τ a bijection from the leaves of T to the vertices of G.