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The problem of computing the chromatic number of a P 5 -free graph (a graph which contains no path on 5 vertices as an induced subgraph) is known to be NP-hard. However, we show that for every fixed integer k, there exists a polynomial-time algorithm determining whether or not a P 5 -free graph admits a k-coloring, and finding one, if it does.

a b s t r a c tWhether the clique-width of graphs in a certain class of graphs is bounded or not, is an important question from an algorithmic point of view, as many problems that are NPhard in general admit polynomial-time solutions when restricted to graphs of bounded clique-width. Over the last few years, many classes of graphs have been shown to have bounded clique-width. For many others, this parameter has been proved to be unbounded. This paper provides a survey of recent results addressing this problem.

The class of fork-free graphs is an extension of claw-free graphs and their subclass of line graphs. The first polynomial-time solution to the maximum weight independent set problem in the class of line graphs, which is equivalent to the maximum matching problem in general graphs, has been proposed by Edmonds in 1965 and then extended to the entire class of claw-free graphs by Minty in 1980. Recently, Alekseev proposed a solution for the larger class of fork-free graphs, but only for the unweighted version of the problem, i.e., finding an independent set of maximum cardinality. In the present paper, we describe the first polynomial-time algorithm to solve the problem for weighted fork-free graphs.

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Abstract. Independent sets, induced matchings and cliques are examples of regular induced subgraphs in a graph. In this paper, we prove that finding a maximum cardinality k-regular induced subgraph is an NP-hard problem for any value of k. We propose a convex quadratic upper bound on the size of a k-regular induced subgraph and characterize those graphs for which this bound is attained. Finally, we extend the Hoffman bound on the size of a maximum 0-regular subgraph (the independence number) from k = 0 to larger values of k.