Let G = (V, E) be a graph on n vertices and R be a set of pairs of vertices in V called requests. A multicut is a subset F of E such that every request xy of R is cut by F , i.e. every xy-path of G intersects F . We show that there exists an O(f (k)n c ) algorithm which decides if there exists a multicut of size at most k. In other words, the MULTICUT problem parameterized by the solution size k is Fixed-Parameter Tractable. General TermsAlgorithms

International audienceWe define NLC Fk to be the restriction of the class of graphs NLC k , where relabelling functions are exclusively taken from a set F of functions from {1,...,k} into {1,...,k}. We characterize the sets of functions F for which NLC Fk is well-quasi-ordered by the induced subgraph relation ≤ i . Precisely, these sets F are those which satisfy that for every f,g∈F , we have Im(f ∘ g) = Im(f) or Im(g ∘ f) = Im(g). To obtain this, we show that words (or trees) on F are well-quasi-ordered by a relation slightly more constrained than the usual subword (or subtree) relation. A class of graphs is n-well-quasi-ordered if the collection of its vertex-labellings into n colors forms a well-quasi-order under ≤ i , where ≤ i respects labels. Pouzet (C R Acad Sci, Paris Sér A-B 274:1677-1680, 1972) conjectured that a 2-well-quasi-ordered class closed under induced subgraph is in fact n-well-quasi-ordered for every n. A possible approach would be to characterize the 2-well-quasi-ordered classes of graphs. In this respect, we conjecture that such a class is always included in some well-quasi-ordered NLC Fk for some family F . This would imply Pouzet's conjecture

International audienceThe Rooted Maximum Leaf problem consists in finding a spanning directed tree rooted at some prescribed vertex of a digraph with the maximum number of leaves. Its parameterized version asks if there exists such a tree with at least $k$ leaves. We use the notion of $s-t$ numbering studied in \cite{stnum, stnumdir,LLWemb} to exhibit combinatorial bounds on the existence of spanning directed trees with many leaves. These combinatorial bounds allow us to produce a constant factor approximation algorithm for finding directed trees with many leaves, whereas the best known approximation algorithm has a $\sqrt{OPT}$-factor \cite{DrescherMaxLeaf}. We also show that Rooted Maximum Leaf admits an edge-quadratic kernel, improving over the vertex-cubic kernel given by Fernau et al \cite{FernauMaxLeaf}

A subgraph T of a digraph D is an out-branching if T is an oriented spanning tree with only one vertex of in-degree zero (called the root). The vertices of T of out-degree zero are leaves. In the Directed Max Leaf problem, we wish to find the maximum number of leaves in an out-branching of a given digraph D (or, to report that D has no out-branching). In the Directed k-Leaf problem, we are given a digraph D and an integral parameter k, and we are to decide whether D has an out-branching with at least k leaves. Recently, Kneis et al. (2008) obtained an algorithm for Directed k-Leaf of running time 4 k · n O (1) . We describe a new algorithm for Directed k-Leaf of running time 3.72 k · n O (1) . This algorithms leads to an O (1.9973 n )-time algorithm for solving Directed Max Leaf on a digraph of order n. The latter algorithm is the first algorithm of running time O (γ n ) for Directed Max Leaf, where γ < 2. In the Rooted Directed k-Leaf problem, apart from D and k, we are given a vertex r of D and we are to decide whether D has an out-branching rooted at r with at least k leaves. Very recently, Fernau et al. (2008) found an O (k 3 )-size kernel for Rooted Directed k-Leaf. In this paper, we obtain an O (k) kernel for Rooted Directed k-Leaf restricted to acyclic digraphs.