Aurélie Lagoutte

Flooding games on graphs

We study a discrete diffusion process introduced in some combinatorial puzzles called Flood-It, Mad Virus, or Honey-Bee, that can be played online and whose computational complexities have recently been studied. Originally defined on regular boards, we show that studying their dynamics directly on general graphs is valuable: we synthesize and extend previous results, we show how to solve Flood-It on cycles by computing a poset height and how to solve the 2-Free-Flood-It variant by computing a graph radius.

IDENTIFYING CODES IN HEREDITARY CLASSES OF GRAPHS AND VC-DIMENSION ∗

An identifying code of a graph is a subset of its vertices such that every vertex of the graph is uniquely identified by the set of its neighbours within the code. We show a dichotomy for the size of the smallest identifying code in classes of graphs closed under induced subgraphs. Our dichotomy is derived from the VC-dimension of the considered class C, that is the maximum VC-dimension over the hypergraphs formed by the closed neighbourhoods of elements of C. We show that hereditary classes with infinite VC-dimension have infinitely many graphs with an identifying code of size logarithmic in the number of vertices while classes with finite VC-dimension have a polynomial lower bound. We then turn to approximation algorithms. We show that the problem of finding a smallest identifying code in a given graph from some class is log-APX-hard for any hereditary class of infinite VC-dimension. For hereditary classes of finite VC-dimension, the only known previous results show that we can approximate the identifying code problem within a constant factor in some particular classes, e.g. line graphs, planar graphs and unit interval graphs. We prove that it can be approximate within a factor 6 for interval graphs. In contrast, we show that on C_4-free bipartite graphs (a class of finite VC-dimension) it cannot be approximated to within a factor of c.log(|V|) for some c>0.

Clique versus Independent Set

Abstract Yannakakis’ Clique versus Independent Set problem (CL–IS) in communication complexity asks for the minimum number of cuts separating cliques from stable sets in a graph, called CS-separator. Yannakakis provides a quasi-polynomial CS-separator, i.e. of size O ( n log n ) , and addresses the problem of finding a polynomial CS-separator. This question is still open even for perfect graphs. We show that a polynomial CS-separator almost surely exists for random graphs. Besides, if H is a split graph (i.e. has a vertex-partition into a clique and a stable set) then there exists a constant c H for which we find a O ( n c H ) CS-separator on the class of H -free graphs. This generalizes a result of Yannakakis on comparability graphs. We also provide a O ( n c k ) CS-separator on the class of graphs without induced path of length k and its complement. Observe that on one side, c H is of order O ( | H | log | H | ) resulting from Vapnik–Chervonenkis dimension, and on the other side, c k is a tower function, due to an application of the regularity lemma. One of the main reason why Yannakakis’ CL–IS problem is fascinating is that it admits equivalent formulations. Our main result in this respect is to show that a polynomial CS-separator is equivalent to the polynomial Alon–Saks–Seymour Conjecture, asserting that if a graph has an edge-partition into k complete bipartite graphs, then its chromatic number is polynomially bounded in terms of k . We also show that the classical approach to the stubborn problem (arising in CSP) which consists in covering the set of all solutions by O ( n log n ) instances of 2-SAT is again equivalent to the existence of a polynomial CS-separator.

The Erdős-Hajnal conjecture for paths and antipaths

We prove that for every k, there exists c k 0 such that every graph G on n vertices with no induced path P k or its complement P k ? contains a clique or a stable set of size n c k .

Strong edge-coloring of ( 3 , Δ ) -bipartite graphs

A strong edge-coloring of a graph G is an assignment of colors to edges such that every color class induces a matching. We here focus on bipartite graphs whose one part is of maximum degree at most?3 and the other part is of maximum degree Δ . For every such graph, we prove that a strong 4 Δ -edge-coloring can always be obtained. Together with a result of Steger and Yu, this result confirms a conjecture of Faudree, Gyarfas, Schelp and Tuza for this class of graphs.

Decomposition techniques applied to the Clique-Stable set separation problem

Abstract In a graph, a Clique-Stable Set separator (CS-separator) is a family C of cuts (bipartitions of the vertex set) such that for every clique K and every stable set S with K ∩ S = ∅ , there exists a cut ( W , W ′ ) in C such that K ⊆ W and S ⊆ W ′ . Starting from a question concerning extended formulations of the Stable Set polytope and a related complexity communication problem, Yannakakis (Yannakakis, 1991) asked in 1991 the following questions: does every graph admit a polynomial-size CS-separator? If not, does every perfect graph do? Several positive and negative results related to this question were given recently. Here we show how graph decomposition can be used to prove that a class of graphs admits a polynomial CS-separator. We apply this method to apple-free graphs and cap-free graphs.

Graph coloring, communication complexity and the stubborn problem (Invited talk).

We discuss three equivalent forms of the same problem arising in communication complexity, constraint satisfaction problems, and graph coloring. Some partial results are discussed.

Alternative method for Hamilton-Jacobi PDEs in image processing

Multiscale signal analysis has been used since the early 1990s as a powerful tool for image processing, notably in the linear case. However, nonlinear PDEs and associated nonlinear operators have advantages over linear operators, notably preserving important features such as edges in images. In this paper, we focus on nonlinear Hamilton-Jacobi PDEs defined with adaptive speeds or, alternatively, on adaptive morphological fiters also called semi-flat morphological operators. Semi-flat morphology were instroduced by H. Heijmans and studied only in the case where the speed (or equivalently the filtering parameter) is a decreasing function of the luminance. It is proposed to extend the definition suggested by H. Heijmans in the case of non decreasing speeds. We also prove that a central property for defining morphological filters, that is the adjunction property, is preserved while dealing with our extended definitions. Finally experimental applications are presented on actual images, including connection of thin lines by semi-flat dilations and image filtering by semi-flat openings.