Stéphan Thomassé

Learn More
In this paper, we give evidence for the problems Disjoint Cycles and Disjoint Paths that they cannot be preprocessed in polynomial time such that resulting instances always have a size bounded by a polynomial in a specified parameter (or, in short: do not have a polynomial kernel); these results are assuming the validity of certain complexity theoretic(More)
We prove that given an undirected graph <i>G</i> on <i>n</i> vertices and an integer <i>k</i>, one can compute, in polynomial time in <i>n</i>, a graph <i>G&#8242;</i> with at most 4<i>k</i><sup>2</sup> vertices and an integer <i>k&#8242;</i> such that <i>G</i> has a feedback vertex set of size at most <i>k</i> iff <i>G&#8242;</i> has a feedback vertex set(More)
We prove that every graph G has a vertex partition into a cycle and an anticycle (a cycle in the complement of G). Emptyset, singletons and edges are considered as cycles. This problem was posed by Lehel and shown to be true for very large graphs by à Luczak, Rödl and Szemerédi [7], and more recently for large graphs by Allen [1]. Many questions deal with(More)
We prove that given an undirected graph G on n vertices and an integer k, one can compute in polynomial time in n a graph G′ with at most 5k+k vertices and an integer k′ such that G has a feedback vertex set of size at most k iff G′ has a feedback vertex set of size at most k′. This result improves a previous O(k) kernel of Burrage et al. [6], and a more(More)
We give a short constructive proof of a theorem of Fisher: every tournament contains a vertex whose second outneighborhood is as large as its ®rst outneighborhood. Moreover, we exhibit two such vertices provided that the tournament has no dominated vertex. The proof makes use of median orders. A second application of median orders is that every tournament(More)
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<sup>c</sup>) algorithm which decides if there exists a multicut of size at most k. In other words, the Multicut(More)
In this paper, we introduce a new technique to give evidence for combinatorial problems that they cannot be preprocessed in polynomial time such that resulting instances always have a size bounded by a polynomial in a specified parameter (or, in short: do not have a polynomial kernel); these results are assuming the validity of certain complexity theoretic(More)
The main result of this paper is that every 4-uniform hypergraph on n vertices and m edges has a transversal with no more than (5n+4m)/21 vertices. In the particular case n = m, the transversal has at most 3n/7 vertices, and this bound is sharp in the complement of the Fano plane. Chvátal and McDiarmid [5] proved that every 3-uniform hypergraph with n(More)