Frédéric Maffray

Completely separable graphs

Abstract We define a property of Boolean functions called separability, and specialize it for a class of functions naturally associated with graphs. “Completely separable graphs” are then derived and characterized in particular by the existence of two crossing chords in any cycle of length at least five. This implies that completely separable graphs are perfect. We present linear time algorithms for the recognition and for the usual optimization problems (maximum weighted stable set and maximum weighted clique).

On the NP-completeness of the k -colorability problem for triangle-free graphs

Abstract We show that the question “Is a graph 3-colorable?” remains NP-complete when restricted to the class of triangle-free graphs with maximum degree 4. Likewise the question “Is a triangle-free graph k -colorable?” is shown to be NP-complete for any fixed value of k ⩾ 4.

Optimizing weakly triangulated graphs

A graph is weakly triangulated if neither the graph nor its complement contains a chordless cycle with five or more vertices as an induced subgraph. We use a new characterization of weakly triangulated graphs to solve certain optimization problems for these graphs. Specifically, an algorithm which runs inO((n + e)n3) time is presented which solves the maximum clique and minimum colouring problems for weakly triangulated graphs; performing the algorithm on the complement gives a solution to the maximum stable set and minimum clique covering problems. Also, anO((n + e)n4) time algorithm is presented which solves the weighted versions of these problems.

Kernels in perfect line-graphs

Abstract A kernel of a directed graph D is a set of vertices which is both independent and absorbant. In 1983, Berge and Duchet conjectured that an undirected graph G is perfect if and only if the following condition is satisfied: “If D is any orientation of G such that every clique of D has a kernel, then D has a kernel.” We prove here that the conjecture holds when G is the line-graph of another graph H , i.e., G represents the incidence between the edges of H .

On the b-Coloring of Cographs and P 4 -Sparse Graphs

A b-coloring of a graph is a coloring such that every color class admits a vertex adjacent to at least one vertex receiving each of the colors not assigned to it. The b-chromatic number of a graph G, denoted by χb(G), is the maximum number t such that G admits a b-coloring with t colors. A graph G is b-continuous if it admits a b-coloring with t colors, for every

Coloring the hypergraph of maximal cliques of a graph with no long path

t = \chi(G), \ldots, \chi_b(G)

On b-colorings in regular graphs

. We define a graph G to be b-monotonic if χb(H1) ≥ χb(H2) for every induced subgraph H1 of G, and every induced subgraph H2 of H1. In this work, we prove that P4-sparse graphs (and, in particular, cographs) are b-continuous and b-monotonic. Besides, we describe a dynamic programming algorithm to compute the b-chromatic number in polynomial time within these graph classes.

Linear recognition of pseudo-split graphs

It is known that all claw-free perfect graphs can be decomposed via clique-cutsets into two types of indecomposable graphs respectively called elementary and peculiar (1988, V. Chvatal and N. Sbihi,J. Combin. Theory Ser. B44, 154?176). We show here that every elementary graph is made up in a well-defined way of a line-graph of bipartite graph and some local augments consisting of complements of bipartite graphs. This yields a complete description of the structure of claw-free Berge graphs and a new proof of their perfectness.