View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Discrete Mathematics 311 (2011) 1670–1680 Contents lists available at ScienceDirect Discrete Mathematics journal homepage: www.elsevier.com/locate/disc On the complexity of the independent set problem in triangle graphs Yury Orlovich a,∗, Jacek Blazewicz b, Alexandre Dolgui c, Gerd Finke d, Valery Gordon e a Department of Discrete Mathematics and Algorithmics, Faculty of Applied Mathematics and Computer Science, Belarusian State University, Nezavisimosti Ave. 4, 220030 Minsk, Belarus b Institute of Computing Science, Poznan University of Technology, ul. Piotrowo 3a, Poznan, Poland c Ecole Nationale Supérieure des Mines de Saint-Etienne, Centre for Industrial Engineering and Computer Science, 158 cours Fauriel, F-42023 Saint Etienne Cedex 2, France d Laboratoire G-SCOP, Université Joseph Fourier, 46 Avenue Félix Viallet, 38031 Grenoble Cedex, France e United Institute of Informatics Problems, National Academy of Sciences of Belarus, 6 Surganova Str., 220012 Minsk, Belarus article info a b s t r a c t Article history: We consider the complexity of the maximum (maximum weight) independent set problem Received 29 October 2009 within triangle graphs, i.e., graphs G satisfying the following triangle condition: for every Received in revised form 31 March 2011 maximal independent set I in G and every edge uv in G − I, there is a vertex w 2 I Accepted 2 April 2011 such that fu; v; wg is a triangle in G. We also introduce a new graph parameter (the Available online 27 April 2011 upper independent neighborhood number) and the corresponding upper independent neighborhood set problem. We show that for triangle graphs the new parameter is equal Keywords: to the independence number. We prove that the problems under consideration are NP- Independent set Neighborhood set complete, even for some restricted subclasses of triangle graphs, and provide several Triangle condition polynomially solvable cases for these problems within triangle graphs. Furthermore, we Triangle graph show that, for general triangle graphs, the maximum independent set problem and the NP-completeness upper independent neighborhood set problem cannot be polynomially approximated Approximability within any fixed constant factor greater than one unless P D NP. ' 2011 Elsevier B.V. All rights reserved. 1. Introduction and some preliminary results In this paper we consider the complexity of the maximum (maximum weight) independent set problem for graphs that satisfy the triangle condition. These problems are well known, and they arise naturally in many fields. Examples for applications of the problems can be found in determining an upper bound on the number of processors needed for parallel execution [24], in map labeling [45], in interval selection problems arising in manufacturing [43], and in scheduling jobs within time windows on parallel machines [14]. We consider only finite undirected simple graphs and use standard graph-theoretic terminologies; see, for example, [6]. For notions related to computational complexity, we follow [15], while for concepts related to approximability, we follow [4]. Let G be a graph with the vertex set V D V .G/ and the edge set E D E.G/. For a subset of vertices X ⊆ V , the subgraph of G induced by X is denoted by G.X/, and G − X D G.V n X/. As usual, NG.x/ denotes the neighborhood of a vertex x 2 V , i.e., the set of all vertices that are adjacent to x in G. The degree of x is defined as degG x D jNG.x/j. The maximum degree of vertices in G is denoted by ∆.G/. The closed neighborhood of x is NGTxUD NG.x/ [ fxg. The proper neighborhood of an edge ∗ Corresponding author. Tel.: +375 17 209 54 71. E-mail addresses: [email protected] (Y. Orlovich), [email protected] (J. Blazewicz), [email protected] (A. Dolgui), [email protected] (G. Finke), [email protected] (V. Gordon). 0012-365X/$ – see front matter ' 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.disc.2011.04.001 Y. Orlovich et al. / Discrete Mathematics 311 (2011) 1670–1680 1671 Fig. 1. Example of triangle (a) and non-triangle (b) graphs. e D xy 2 E is defined as PNG.e/ D NGTxU\ NGTyU; i.e., the set PNG.e/ consists of x, y, and all vertices adjacent to both x and y. ⊆ D S T UD [ If X V then NG.X/ x2X NG.x/ and NG X NG.X/ X. The subscript G will be omitted whenever the context is clear. The notation u ∼ v (u 6∼ v, respectively) means that the vertices u and v are adjacent (non-adjacent, respectively). For disjoint sets of vertices U and W , the notation U ∼ W (U 6∼ W , respectively) means that every vertex of U is adjacent (non-adjacent, respectively) to every vertex of W . When U D fug, we also write u ∼ W and u 6∼ W instead of fug ∼ W and fug 6∼ W , respectively. For a set H of graphs, a graph G is called H-free if no induced subgraph of G is isomorphic to a graph in H. The complement G of a graph G is that graph whose vertex set is V .G/ and where uv is an edge of G if and only if uv is not an edge of G. The ∼ notation G1 D G2 means that graph G1 is isomorphic to graph G2. For two (vertex-disjoint) graphs G1 and G2, the (disjoint) union G1 [ G2 of G1 and G2 is the (disconnected) graph with V .G1 [ G2/ D V .G1/ [ V .G2/ and E.G1 [ G2/ D E.G1/ [ E.G2/. For a positive integer n, the disjoint union of n copies of a graph G is denoted by nG. For example, the graph mK2 consists of m pairwise disjoint edges. Km;n denotes the complete bipartite graph with partition classes of cardinalities m and n; Kn is the complete graph on n vertices; Cn and Pn are the chordless cycle and the chordless path on n vertices, respectively. The graph K1;n is also called a star, and K3 D C3 is called a triangle. At the same time, the star K1;3 is known as a claw. K4 − e is a graph obtained from the complete graph K4 by deleting an edge, and K1;3 C e is a graph obtained from the claw K1;3 by adding an edge between any two selected non-adjacent vertices. Let K0 be the null graph; i.e., K0 has no vertices and edges. See, for example, [44]. For a graph G, a subset I ⊆ V .G/ of vertices is called an independent (or stable) set if no two vertices in I are adjacent. The maximum cardinality taken over all independent sets of G is the independence number, and is denoted by α.G/. We say that an independent set I is maximum if jIj D α.G/. An independent set I is maximal if no other independent set contains I. The minimum cardinality over all maximal independent sets of G is called the independent domination number and is denoted by i.G/. A vertex subset U ⊆ V .G/ is called a clique in G if the vertices in U are pairwise adjacent. A clique U is maximal if it is not contained in a larger clique. A weighted graph is a pair .G; w/ consisting of a graph G and a non-negative weight function w: V .G/ ! Z that associates an integer w.v/ ≥ 0 to each vertex v of G (the weight of the vertex v). The weight of a subset of vertices S ⊆ V .G/ is defined P as v2S w.v/ and is denoted by w.S/. The weighted independence number αw.G/ of a weighted graph .G; w/ is the maximum weight of an independent set in G. A graph is called complement reducible (a cograph for short) if, for any induced subgraph F of G with at least two vertices, either F or the complement to F is disconnected. Cographs have a characterization in terms of forbidden induced subgraphs: they are P4-free graphs [13]. A graph G is called a general partition graph [33] if there exists a set L and an assignment of non-empty subsets Lu ⊆ L to the vertices u 2 V .G/ with the following properties: (i) any two vertices u and v are adjacent if and only if Lu \ Lv 6D ;, and (ii) for every maximal independent set I of G, the collection fLx V x 2 Ig partitions L. A graph G is called a triangle graph if it satisfies the following triangle condition: for every maximal independent set I and every edge uv in G − I, there exists a vertex w 2 I such that fu; v; wg induces a triangle in G. In other words, G is a triangle graph if and only if every maximal independent set in G intersects all sets PNG.e/, where e 2 E.G/. An example of a triangle graph is given in Fig. 1(a). In this graph, a maximal independent set consists either of the vertex of degree 4, or two non-adjacent vertices of degree 3. In both cases, it is easy to see that the triangle condition holds. On the other hand, graph G shown in Fig. 1(b) is not triangle, since, for the maximal independent set I D fx; yg, the condition does not hold because of the edge uv 2 E.G − I/. The definition of the triangle condition was introduced by McAvaney et al.