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International audienceIn this paper, we show that the problems Disjoint Cycles and Disjoint Paths do not have polynomial kernels, unless N P ⊆ c o N P / p o l y . Thus, these problems do not allow polynomial time preprocessing that results in instances whose size is bounded by a polynomial in the parameter at hand. We build upon recent results by Bodlaender et al. [6] and Fortnow and Santhanam [20], that show that NP-complete problems that are ‘or-compositional’ do not have polynomial kernels, unless N P ⊆ c o N P / p o l y . To this machinery, we add a notion of transformation, and obtain that Disjoint Cycles, and Disjoint Paths do not have polynomial kernels, unless N P ⊆ c o N P / p o l y . For the proof, we introduce a problem on strings, called Disjoint Factors, and first show that this problem has no polynomial kernel unless N P ⊆ c o N P / p o l y . We also show that the related Disjoint Cycles Packing problem has a kernel of size O ( k log k )

A tournament is a complete graph with its edges directed, and colouring a tournament means partitioning its vertex set into transitive subtournaments. For some tournaments H there exists c such that every tournament not containing H as a subtournament has chromatic number at most c (we call such a tournament H a hero); for instance, all tournaments with at most four vertices are heroes. In this paper we explicitly describe all heroes.

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