Stéphan Thomassé

A 4 k 2 kernel for feedback vertex set

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′</i> with at most 4<i>k</i>² vertices and an integer <i>k′</i> such that <i>G</i> has a feedback vertex set of size at most <i>k</i> iff <i>G′</i> has a feedback vertex set of size at most <i>k′</i>. This result improves a previous <i>O</i>(<i>k</i>¹¹) kernel of Burrage et al., and a more recent cubic kernel of Bodlaender. This problem was communicated by Fellows.

The Minimum Feedback Arc Set Problem is NP-Hard for Tournaments

Answering a question of Bang-Jensen and Thomassen [4], we prove that the minimum feedback arc set problem is NP-hard for tournaments.

Kernel Bounds for Disjoint Cycles and Disjoint Paths

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 assumptions. We build upon recent results by Bodlaender et al. [3] and Fortnow and Santhanam [13], that show that NP-complete problems that are or-compositional do not have polynomial kernels, unless NP ⊆ coNP/poly. To this machinery, we add a notion of transformation, and thus obtain that Disjoint Cycles and Disjoint Paths do not have polynomial kernels, unless NP ⊆ coNP/poly. We also show that the related Disjoint Cycles Packing problem has a kernel of size O(k logk).

Total domination of graphs and small transversals of hypergraphs

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 vertices and edges has a transversal of size n/2. Two direct corollaries of these results are that every graph with minimal degree at least 3 has total domination number at most n/2 and every graph with minimal degree at least 4 has total domination number at most 3n/7. These two bounds are sharp.

Multicut is FPT

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)nc) 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.

The Domination Number of Grids

In this paper, we conclude the calculation of the domination number of all (n,m) grid graphs. Indeed, we prove Chang’s conjecture saying that for every 16≤n≤m, γ(Gn,m)=⌊(n+2)(m+2)5⌋-4.

Median orders of tournaments: A tool for the second neighborhood problem and Sumner's conjecture

We prove the following recent conjecture of Halin. Let Γ0 be the class of all graphs, and for every ordinal μ > 0 let Γμ be the class of all graphs containing infinitely many disjoint connected graphs from Γλ, for every λ < μ. Then a graph lies in all these classes Γμ if and only if it contains a subdivision of the infinite binary tree. Published by John Wiley & Sons, Inc., 2000 J Graph Theory 35: 273–277, 2000We give a short constructive proof of a theorem of Fisher: every tournament contains a vertex whose second outneighborhood is as large as its first 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 of order 2n - 2 contains every arborescence of order n > 1. This is a particular case of Sumners conjecture: every tournament of order 2n - 2 contains every oriented tree of order n > 1. Using our method, we prove that every tournament of order (7n - 5)-2 contains every oriented tree of order n.

Partitioning a graph into a cycle and an anticycle, a proof of Lehel's conjecture

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 Luczak, Rodl and Szemeredi [T. Luczak, V. Rodl, E. Szemeredi, Partitioning two-colored complete graphs into two monochromatic cycles, Combin. Probab. Comput. 7 (1998) 423-436], and more recently for large graphs by Allen [P. Allen, Covering two-edge-coloured complete graphs with two disjoint monochromatic cycles, Combin. Probab. Comput. 17 (2008) 471-486].

Oriented Hamiltonian Paths in Tournaments

We prove that with three exceptions, every tournament of order n contains each oriented path of order n. The exceptions are the antidirected paths in the 3-cycle, in the regular tournament on 5 vertices, and in the Paley tournament on 7 vertices.

Simultaneously Satisfying Linear Equations Over F_2: MaxLin2 and Max-r-Lin2 Parameterized Above Average.

In the parameterized problem MaxLin2-AA[