View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Theoretical Computer Science 412 (2011) 2496–2502 Contents lists available at ScienceDirect Theoretical Computer Science journal homepage: www.elsevier.com/locate/tcs Block-graph widthI Maw-Shang Chang a, Ling-Ju Hung a, Ton Kloks, Sheng-Lung Peng b,∗ a Department of Computer Science and Information Engineering, National Chung Cheng University, Chiayi 62107, Taiwan b Department of Computer Science and Information Engineering, National Dong Hwa University, Hualien 97401, Taiwan article info a b s t r a c t Keywords: Let G be a class of graphs. A graph G has G-width k if there are k independent sets N1;:::; Nk Probe graphs in G such that G can be embedded into a graph H 2 G such that for every edge e in H which Parameterized algorithms is not an edge in G, there exists an i such that both endvertices of e are in . For the class Block graphs Ni B of block graphs we show that graphs with B-width at most 4 are perfect. We also show that B-width is NP-complete and show that it is fixed-parameter tractable. For the class C of complete graphs, similar results are also obtained. ' 2010 Elsevier B.V. All rights reserved. 1. Introduction Let G D .V ; E/ be a finite, simple, and undirected graph in which V and E are the sets of vertices and edges of G, respectively. Let G be a class of graphs. A graph G is a probe graph of G (or equivalently, probe G-graph) if its vertices can be partitioned into a set P and an independent set N such that G can be embedded into a graph H 2 G by adding edges between certain vertices of N. The vertices of P are called probes and those of N are called nonprobes. The graph H is called an embedding of G. It is well known that graphs provide an important representation for modeling data (or information). Assume that G D .P C N; E/ is a probe G-graph. The relations between vertices in N are missed. The recognition problem of probe G-graph for graph G is to ask whether we can add edges between vertices of N such that the resulting graph is a G graph. The recognition problem of probe interval graphs was introduced by Zhang et al. [14,21]. This problem stems from the physical mapping of chromosomal DNA of humans and other species. Since then probe graphs of many other graph classes have been investigated by various authors [1,3,4,13]. Since the relations between vertices in N are missed, vertices in N can be treated as polluted (e.g., by a drop of ink). In this paper we generalize the concept of traditional probe graphs to k-probe graphs. That is, graph G is polluted by k drops of ink. We define the following width parameter of graph classes. Definition 1. Let G be a class of graphs. 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 2 G of G such that for every edge e D .x; y/ in H which is not an edge of G there exists an i with x; y 2 Ni. In the case that k D 1, G is a probe G-graph. The collection of independent sets Ni, i D 1;:::; k for which there exists an embedding H as stipulated in Definition 1 is called a witness of H. The G-width problem for G is to determine the G-width of G. We refer to the partitioned case of the problem when a collection of independent sets Ni, i D 1;:::; k is a part of the input; otherwise, it is an unpartitioned case. D n Sk Sk For historical reasons we call the set of vertices P V iD1 Ni the set of probes and the vertices of iD1 Ni the set of nonprobes. I A preliminary version of this paper appears in the Proceedings of TAMC 2009, LNCS, vol. 5532, 2009, pp. 150–157. ∗ Corresponding author. E-mail addresses: [email protected] (M.-S. Chang), [email protected] (L.-J. Hung), [email protected] (S.-L. Peng). 0304-3975/$ – see front matter ' 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.tcs.2010.10.041 M.-S. Chang et al. / Theoretical Computer Science 412 (2011) 2496–2502 2497 r r ✟r ✟ r r r @ r r A ✁@ r @ H A ✁ ✟@ @@ r r r HHA✟✁r ✟ Fig. 1. A diamond, a hole, and a gem. A graph has rankwidth at most 1 if and only if it has no vertex minor isomorphic to the gem [16]. In this paper we investigate the width-parameter for the class B (respectively, C) of block (respectively, complete) graphs, henceforth called the block-graph width (respectively, complete-graph width), or B-width (respectively, C-width). If a graph G has B-width k then we call G also a k-probe block graph. The rest of the paper is organized as follows: In Section 2, we present some definitions and notations for this paper. In Section 3, we present some properties for k-probe block (and complete) graphs. In Section 4, we propose polynomial- time algorithms for recognizing graphs of B-width (and C-width) at most k. Both partitioned and unpartitioned cases are considered. For the unpartitioned case, our algorithm is fixed-parameter with respect to the width k. In Section 5, we show that both B-Width and C-Width are NP-complete. In Section 6, we show that the class of graphs with complete-width at most k has a finite obstruction set. Finally, some concluding remarks are given in the last section. 2. Definitions and notation Let G D .V ; E/ be a finite, simple, and undirected graph in which V and E are the sets of vertices and edges of G, respectively. Let n D jV j and m D jEj. We denote edges of G D .V ; E/ as .x; y/ and we call x and y the endvertices of the edge. Let G be the complement of G, i.e., G D .V ; E/, where E D f.x; y/2 = E j x; y 2 V g. A graph G D .V ; E/ is connected if for any two vertices x and y in V there exists a path from x to y in G.A component of G is a maximal connected subgraph of G. A vertex x in G is a cutvertex if the removal of x from G results a disconnect graph. A component is biconnected if it does not contain any cutvertex. A biconnected component is also called a block.A hole in G is an induced subgraph isomorphic to a cycle of length at least 4. For a subset W ⊆ V , we write GTW U for the subgraph of G induced by W . A vertex subset W is an independent set (respectively, clique) of G if the edge set of GTW U is empty (respectively, complete, i.e., any two vertices have an edge). For a vertex x, let N.x/ D fv 2 V j .x; v/ 2 Eg and NTxUD N.x/ [ fxg. We use NW .x/ (respectively, NW TxU) to denote N.x/ \ W (respectively, NTxU\ W ). A vertex x is universal if NTxUD V and it is isolated if N.x/ D ?. For convenience, we use A C B (respectively, A − B) instead of A [ B (respectively, A n B) for two sets A and B. In the case that B contains only one element, e.g., B D fxg, we use A C x (respectively, A − x) instead of A C B (respectively, A − B). Further, for a vertex set W , we write G − W to denote the subgraph GTV − W U. For a vertex x we write G − x rather than GTV − fxgU. For an edge e we use G − e to denote the subgraph .V ; E − e/. Block graphs were introduced in [12]. They are special ptolemaic graphs, i.e., gem-free chordal graphs, in which every biconnected component is a clique. We adapt the definition as it is presented in [2, Definition 10.2.3], to allow for disconnected graphs (Fig. 1). 3. k-probe block graphs In this section we show some properties for k-probe block graphs. We first consider the block graphs. Definition 2 ([2,12]). A graph G is a block graph if every biconnected component, or `block,' is complete. Equivalently, G is a block graphs if G is chordal and has no induced subgraph isomorphic to K4 − e, that is, the diamond. Definition 2 focuses on biconnected components. In the following, we propose an alternative definition for block graphs based only on connected components. Theorem 3. A graph is a block graph if and only if every connected induced subgraph is a clique or else has a cutvertex. Proof. Let G D .V ; E/ be a block graph and let GTW U be a subgraph induced by W ⊆ V . Assume GTW U is connected. Then GTW U is a block graph, since by Definition 2, `to be a block graph' is a hereditary property. If GTW U is not complete then it has a cutvertex by Definition 2. Assume every connected induced subgraph of a graph G has a cutvertex or is a clique.