An algorithm for finding homogeneous pairs

AbstractA homogeneous pair in a graph G = (V, E) is a pair Q1, Q2 of disjoint sets of vertices in this graph such that every vertex of V (Q1 ∪ Q2) is adjacent either to all vertices of Q1 or to none of the vertices of Q1 and is adjacent either to all vertices of Q2 or to none of the vertices of Q2. Also ¦Q1¦ ⩾ 2 or ¦Q2¦⩾ 2 and ¦V (Q1 ∪ Q2)¦ ⩾ 2. In this paper we present an O(mn3)-time algorithm which determines whether a graph contains a homogeneous pair, and if possible finds one

The world of hereditary graph classes viewed through Truemper configurations

In 1982 Truemper gave a theorem that characterizes graphs whose edges can be labeled so that all chordless cycles have prescribed parities. The characterization states that this can be done for a graph G if and only if it can be done for all induced subgraphs of G that are of few speci c types, that we will call Truemper con gurations. Truemper was originally motivated by the problem of obtaining a co-NP characterization of bipartite graphs that are signable to be balanced (i.e. bipartite graphs whose node-node incidence matrices are balanceable matrices). The con gurations that Truemper identi ed in his theorem ended up playing a key role in understanding the structure of several seemingly diverse classes of objects, such as regular matroids, balanceable matrices and perfect graphs. In this survey we view all these classes, and more, through the excluded Truemper con gurations, focusing on the algorithmic consequences, trying to understand what structurally enables e cient recognition and optimization algorithms

Algorithmic Aspects of Switch Cographs

This paper introduces the notion of involution module, the first generalization of the modular decomposition of 2-structure which has a unique linear-sized decomposition tree. We derive an O(n^2) decomposition algorithm and we take advantage of the involution modular decomposition tree to state several algorithmic results. Cographs are the graphs that are totally decomposable w.r.t modular decomposition. In a similar way, we introduce the class of switch cographs, the class of graphs that are totally decomposable w.r.t involution modular decomposition. This class generalizes the class of cographs and is exactly the class of (Bull, Gem, Co-Gem, C_5)-free graphs. We use our new decomposition tool to design three practical algorithms for the maximum cut, vertex cover and vertex separator problems. The complexity of these problems was still unknown for this class of graphs. This paper also improves the complexity of the maximum clique, the maximum independant set, the chromatic number and the maximum clique cover problems by giving efficient algorithms, thanks to the decomposition tree. Eventually, we show that this class of graphs has Clique-Width at most 4 and that a Clique-Width expression can be computed in linear time

The stable set polytope of claw-free graphs with stability number at least four. I. Fuzzy antihat graphs are W-perfect

Abstract Fuzzy antihat graphs are graphs obtained as 2-clique-bond compositions of fuzzy line graphs with three different types of three-cliqued graphs. By the decomposition theorem of Chudnovsky and Seymour [2] , fuzzy antihat graphs form a large subclass of claw-free, not quasi-line graphs with stability number at least four and with no 1-joins. A graph is W -perfect if its stable set polytope is described by: nonnegativity, rank, and lifted 5-wheel inequalities. By exploiting the polyhedral properties of the 2-clique-bond composition, we prove that fuzzy antihat graphs are W -perfect and we move a crucial step towards the solution of the longstanding open question of finding an explicit linear description of the stable set polytope of claw-free graphs

Separability and Vertex Ordering of Graphs

Many graph optimization problems, such as finding an optimal coloring, or a largest clique, can be solved by a divide-and-conquer approach. One such well-known technique is decomposition by clique separators where a graph is decomposed into special induced subgraphs along their clique separators. While the most common practice of this method employs minimal clique separators, in this work we study other variations as well. We strive to characterize their structure and in particular the bound on the number of atoms. In fact, we strengthen the known bounds for the general clique cutset decomposition and the minimal clique separator decomposition. Graph ordering is the arrangement of a graph’s vertices according to a certain logic and is a useful tool in optimization problems. Special types of vertices are often recognized in graph classes, for instance it is well-known every chordal graph contains a simplicial vertex. Vertex-ordering, based on such properties, have originated many linear time algorithms. We propose to define a new family named SE-Class such that every graph belonging to this family inherently contains a simplicial extreme, that is a vertex which is either simplicial or has exactly two neighbors which are non-adjacent. Our family lends itself to an ordering based on simplicial extreme vertices (named SEO) which we demonstrate to be advantageous for the coloring and maximum clique problems. In addition, we examine the relation of SE-Class to the family of (Even-Hole, Kite)-free graphs and show a linear time generation of SEO for (Even-Hole, Diamond, Claw)-free graphs. We showcase the applications of those two core tools, namely clique-based decomposition and vertex ordering, on the (Even-Hole, Kite)-free family