A polynomial Turing-kernel for weighted independent set in bull-free graphs

The maximum stable set problem is NP-hard, even when restricted to triangle-free graphs. In particular, one cannot expect a polynomial time algorithm deciding if a bull-free graph has a stable set of size k, when k is part of the instance. Our main result in this paper is to show the existence of an FPT algorithm when we parameterize the problem by the solution size k. A polynomial kernel is unlikely to exist for this problem. We show however that our problem has a polynomial size Turingkernel. More precisely, the hard cases are instances of size O(k5). As a byproduct, if we forbid odd holes in addition to the bull, we show the existence of a polynomial time algorithm for the stable set problem. We also prove that the chromatic number of a bull-free graph is bounded by a function of its clique number and the maximum chromatic number of its triangle-free induced subgraphs. All our results rely on a decomposition theorem for bull-free graphs due to Chudnovsky which is modified here, allowing us to provide extreme decompositions, adapted to our computational purpose

EPTAS and Subexponential Algorithm for Maximum Clique on Disk and Unit Ball Graphs

A (unit) disk graph is the intersection graph of closed (unit) disks in the plane. Almost three decades ago, an elegant polynomial-time algorithm was found for Maximum Cliqe on unit disk graphs [Clark, Colbourn, Johnson; Discrete Mathematics ’90]. Since then, it has been an intriguing open question whether or not tractability can be extended to general disk graphs. We show that the disjoint union of two odd cycles is never the complement of a disk graph nor of a unit (3-dimensional) ball graph. From that fact and existing results, we derive a simple QPTAS and a subexponential algorithm running in time 2O˜(n2/3) for Maximum Cliqe on disk and unit ball graphs. We then obtain a randomized EPTAS for computing the independence number on graphs having no disjoint union of two odd cycles as an induced subgraph, bounded VC-dimension, and linear independence number. This, in combination with our structural results, yields a randomized EPTAS for Max Cliqe on disk and unit ball graphs. Max Cliqe on unit ball graphs is equivalent to finding, given a collection of points in R3, a maximum subset of points with diameter at most some fixed value. In stark contrast, Maximum Cliqe on ball graphs and unit 4-dimensional ball graphs, as well as intersection graphs of filled ellipses (even close to unit disks) or filled triangles is unlikely to have such algorithms. Indeed, we show that, for all those problems, there is a constant ratio of approximation which cannot be attained even in time 2n1−ε, unless the Exponential Time Hypothesis fails

Subdivisions in digraphs of large out-degree or large dichromatic number

In 1985, Mader conjectured the existence of a function f such that every digraph with minimum out-degree at least f ( k ) contains a subdivision of the transitive tournament of order k . This conjecture is still completely open, as the existence of f ( 5 ) remains unknown. In this paper, we show that if D is an oriented path, or an in-arborescence (i.e., a tree with all edges oriented towards the root) or the union of two directed paths from x to y and a directed path from y to x , then every digraph with minimum out-degree large enough contains a subdivision of D . Additionally, we study Mader's conjecture considering another graph parameter. The dichromatic number of a digraph D is the smallest integer k such that D can be partitioned into k acyclic subdigraphs. We show that any digraph with dichromatic number greater than 4 m ( n − 1 ) contains every digraph with n vertices and m arcs as a subdivision. We show that any digraph with dichromatic number greater than 4 m ( n − 1 ) contains every digraph with n vertices and m arcs as a subdivision

Tournaments and colouring

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. © 2012 Elsevier Inc

Tournaments and colouring

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. © 2012 Elsevier Inc

Quantification of physiological colour changes in common cuttlefish skin explants as innovative endpoint for contaminant risk assessment

International audienc

An 8.35 Mb overlapping interstitial deletion of 8q24 in two patients with coloboma, congenital heart defect, limb abnormalities, psychomotor retardation and convulsions

Chromosome analysis in two young patients with multiple congenital anomalies revealed a de novo interstitial deletion of 8q that has not been reported before. The deletions were overlapping by 8.35 Mb (8q24.21q24.23). The clinical features shared by our patients were coloboma, VSD, digital abnormalities, congenital dislocation of a hip, feeding problems, psychomotor delay and convulsions. The deletion included the region for Langer-Giedion syndrome (TRPS1 and EXT1) in the girl only. However, she is too young to present features of this syndrome, apart from dysmorphic features like a bulbous nose and notched alae nasi. Several genes are present in the commonly deleted region, including genes with unknown function, and genes for which haploinsufficiency is known to have no phenotypic effect in mice (Wnt1). A gene that might play a role in the convulsions of our patients is KCNQ3