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Characterization and Recognition of P4-Sparse Graphs Partitionable Into K Independent Sets and ℓ Cliques✩

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Characterization and Recognition of P4-Sparse Graphs Partitionable Into K Independent Sets and ℓ Cliques✩ Discrete Applied Mathematics 159 (2011) 165–173 Contents lists available at ScienceDirect Discrete Applied Mathematics journal homepage: www.elsevier.com/locate/dam Characterization and recognition of P4-sparse graphs partitionable into k independent sets and ` cliquesI Raquel S.F. Bravo a, Sulamita Klein a, Loana Tito Nogueira b, Fábio Protti b,∗ a COPPE-PESC, Universidade Federal do Rio de Janeiro, Brazil b IC, Universidade Federal Fluminense, Brazil article info a b s t r a c t Article history: In this work, we focus on the class of P4-sparse graphs, which generalizes the well-known Received 28 December 2009 class of cographs. We consider the problem of verifying whether a P4-sparse graph is a Received in revised form 22 October 2010 .k; `/-graph, that is, a graph that can be partitioned into k independent sets and ` cliques. Accepted 28 October 2010 First, we describe in detail the family of forbidden induced subgraphs for a cograph to be Available online 27 November 2010 a .k; `/-graph. Next, we show that the same forbidden structures suffice to characterize P -sparse graphs which are .k; `/-graphs. Finally, we describe how to recognize .k; `/-P - Keywords: 4 4 sparse graphs in linear time by using special auxiliary cographs. .k; `/-graphs Cographs ' 2010 Elsevier B.V. All rights reserved. P4-sparse graphs 1. Introduction The class of P4-sparse graphs was introduced by Hoàng [14] as the class of graphs for which every set of five vertices induces at most one P4. Hoàng also gave a number of characterizations for these graphs, and showed that P4-sparse graphs are perfect (a graph G is perfect if for every induced subgraph H of G, the chromatic number of H equals the largest number of pairwise adjacent vertices in H). The class of P4-sparse graphs generalizes both the cographs and the P4-reducible graphs. The well-known class of cographs was introduced in the early 1970s by Lerchs [20] as the class of graphs for which no induced subgraph is isomorphic to a P4, and P4-reducible graphs were introduced by Jamison and Olariu [16] as those in which no vertex belongs to more than one induced P4. Both cographs and P4-reducible graphs can be recognized in linear time [4,6,16]. In [17], Jamison and Olariu gave several structural theorems for P4-sparse graphs, including a constructive characterization asserting that P4-sparse graphs are exactly the graphs constructible from single-vertex graphs by three graph operations. This result implies that P4-sparse graphs have a unique tree representation up to isomorphism, which leads to a linear time recognition algorithm for this class. The classes of P4-sparse graphs, cographs and P4-reducible graphs have been studied extensively in recent years and have applications in many areas of applied mathematics, computer science and engineering, mainly because of their good algorithmic and structural properties. The purpose of this paper is to study the partition of the vertex set of P4-sparse graphs into parts which can be independent sets or cliques. The problem of partitioning the vertex set of a graph into k independent sets and ` cliques, for fixed k; `, is known as the .k; `/-partition problem, and is a natural generalization of the coloring problem (where ` D 0) I This work has been partially supported by CNPq, CAPES, and FAPERJ (E-26/110.552/2010). ∗ Corresponding address: IC/UFF - Rua Passo da Pátria, 156, Bloco E, 3o. andar - CEP 24210-240 - Niterói, RJ, Brazil. Tel.: +55 21 26295636; fax: +55 21 26295669. E-mail addresses: [email protected] (R.S.F. Bravo), [email protected] (S. Klein), [email protected] (L.T. Nogueira), [email protected] (F. Protti). 0166-218X/$ – see front matter ' 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.dam.2010.10.019 166 R.S.F. Bravo et al. / Discrete Applied Mathematics 159 (2011) 165–173 and the clique partition problem (where k D 0). Graphs that can be partitioned in this way are called .k; `/-graphs, and were introduced by Brandstädt, in [1]. The matrix partition problem is a yet wider generalization where the parts satisfy not only internal restrictions (such as being an independent set or a clique), but also external restrictions (e.g., being pairwise connected by all possible edges). We refer to [11,12] for details. It is known that, for k ≥ 3 or ` ≥ 3, the problem of recognizing .k; `/-graphs is NP-complete [2]. Due to this fact, many works have considered special families of .k; `/-graphs that can be efficiently recognized; for example, .k; `/-chordal graphs [13], .k; `/-cographs [3,8,10], and .k; `/-perfect graphs [9]. Since P4-sparse graphs have bounded clique width, the problem of deciding whether a P4-sparse graph is a .k; `/-graph for fixed k; ` can be solved in linear time, using a general argument (see [7] for details). We describe a simpler linear time method based on our characterization of P4-sparse graphs that are .k; `/-graphs. 2. Background and terminologies Given a simple graph G D .V ; E/, we denote by G the complement of G. For V 0 ⊆ V ; GTV 0U denotes the subgraph of G induced by V 0.A clique (independent set) is a subset of vertices inducing a complete (edgeless) subgraph, not necessarily maximal. The M-partition problem was introduced by Feder et al. [11], as follows. Let M be a fixed symmetric m × m matrix with entries M.i; j/ 2 f0; 1; ∗}. An M-partition of a graph G is a partition of the vertex set V .G/ into m parts V1; V2;:::; Vm such that Vi is a clique if M.i; i/ D 1, an independent set if M.i; i/ D 0, or with no internal restriction if M.i; i/ D ∗; and such that parts Vi and Vj are completely adjacent if M.i; j/ D 1, completely non-adjacent if M.i; j/ D 0, or with no restriction if M.i; j/ D ∗. Thus the diagonal entries define whether the parts are cliques or independent sets, and the off-diagonal entries define whether the parts are completely adjacent or non-adjacent (with ∗ meaning no restriction). A graph G that does not admit an M-partition is called an M-obstruction.A minimal M-obstruction is a graph G which is an M-obstruction and such that every proper induced subgraph of G admits an M-partition. Given a graph G, sometimes it is useful to associate lists with its vertices. A list M-partition of G with respect to lists L.v/; v 2 V .G/, is an M-partition of G in which each vertex v belongs to a part Vi such that i 2 L.v/. Note that the trivial case, when all lists are L.v/ D f1; 2;:::; mg, corresponds to the situation when no lists are given. A matrix without diagonal ∗'s may be written in a block form, by first listing the rows and columns with diagonal 0's, then those with diagonal 1's. The matrix falls into four blocks, a k by k diagonal matrix A with a diagonal of 0's, an ` by ` diagonal matrix B with a diagonal of 1's, and a k by ` off-diagonal matrix C and its transpose. We say in this case that M is an .A; B; C/-block matrix. We shall say that M is a constant matrix if the off-diagonal entries of A are all the same, say equal to a, the off-diagonal entries of B are all the same, say b, and all entries of C are the same, say c. In this case, we also say that M is an .a; b; c/-block matrix. Observe that if the matrix M is a .∗; ∗; ∗/-block matrix, then an M-partition of G is precisely a partition of the vertices of G into k independent sets and ` cliques. In this case, G is said to be a .k; `/-graph. Feder et al. [10] studied this case (with lists) for the class of cographs. For a .k; `/-graph G, we write V D S1 [···[ Sk [ C1 [···[ C`, where each Sj is an independent set and each Ci is a clique. Such a partition is called a .k; `/-partition of G. In this definition some sets may be empty. The complete (resp. edgeless) graph on r vertices is denoted by Kr (resp. Ir ). Given two graphs G1 D .V1; E1/ and G2 D .V2; E2/, with V1 \ V2 D;, the graph G1 [ G2 (called the union of G1 and G2) is the graph with vertex set V1 [ V2 and edge set E1 [ E2, and the graph G1 C G2 (called the join of G1 and G2) is the graph with vertex set V1 [ V2, and edge set E1 [ E2 [ f.x; y/ j x 2 V1; y 2 V2g.A cograph [5,20,21] is recursively defined as follows: – K1 is a cograph; – if G is a cograph then G is also a cograph; – if G and H are cographs, then G [ H is also a cograph. In [5], Corneil et al. proved that a graph G is a cograph if and only if G contains no induced P4 (a chordless path with four vertices). It follows from the definition of cographs that every cograph G is associated with a unique rooted tree T .G/, called the cotree of G, whose leaves are precisely the vertices of G and whose internal nodes are of two types, 0 or 1, in such a way that two vertices x and y are adjacent in G if and only if their lowest common ancestor in T .G/ is a type-1 node.
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