10 research outputs found
Clique-Stable Set separation in perfect graphs with no balanced skew-partitions
Inspired by a question of Yannakakis on the Vertex Packing polytope of
perfect graphs, we study the Clique-Stable Set Separation in a non-hereditary
subclass of perfect graphs. A cut (B,W) of G (a bipartition of V(G)) separates
a clique K and a stable set S if and . A
Clique-Stable Set Separator is a family of cuts such that for every clique K,
and for every stable set S disjoint from K, there exists a cut in the family
that separates K and S. Given a class of graphs, the question is to know
whether every graph of the class admits a Clique-Stable Set Separator
containing only polynomially many cuts. It is open for the class of all graphs,
and also for perfect graphs, which was Yannakakis' original question. Here we
investigate on perfect graphs with no balanced skew-partition; the balanced
skew-partition was introduced in the proof of the Strong Perfect Graph Theorem.
Recently, Chudnovsky, Trotignon, Trunck and Vuskovic proved that forbidding
this unfriendly decomposition permits to recursively decompose Berge graphs
using 2-join and complement 2-join until reaching a basic graph, and they found
an efficient combinatorial algorithm to color those graphs. We apply their
decomposition result to prove that perfect graphs with no balanced
skew-partition admit a quadratic-size Clique-Stable Set Separator, by taking
advantage of the good behavior of 2-join with respect to this property. We then
generalize this result and prove that the Strong Erdos-Hajnal property holds in
this class, which means that every such graph has a linear-size biclique or
complement biclique. This property does not hold for all perfect graphs (Fox
2006), and moreover when the Strong Erdos-Hajnal property holds in a hereditary
class of graphs, then both the Erdos-Hajnal property and the polynomial
Clique-Stable Set Separation hold.Comment: arXiv admin note: text overlap with arXiv:1308.644
Multigraphs without large bonds are wqo by contraction
We show that the class of multigraphs with at most connected components
and bonds of size at most is well-quasi-ordered by edge contraction for all
positive integers . (A bond is a minimal non-empty edge cut.) We also
characterize canonical antichains for this relation and show that they are
fundamental
Induced minors and well-quasi-ordering
A graph is an induced minor of a graph if it can be obtained from an
induced subgraph of by contracting edges. Otherwise, is said to be
-induced minor-free. Robin Thomas showed that -induced minor-free
graphs are well-quasi-ordered by induced minors [Graphs without and
well-quasi-ordering, Journal of Combinatorial Theory, Series B, 38(3):240 --
247, 1985].
We provide a dichotomy theorem for -induced minor-free graphs and show
that the class of -induced minor-free graphs is well-quasi-ordered by the
induced minor relation if and only if is an induced minor of the gem (the
path on 4 vertices plus a dominating vertex) or of the graph obtained by adding
a vertex of degree 2 to the complete graph on 4 vertices. To this end we proved
two decomposition theorems which are of independent interest.
Similar dichotomy results were previously given for subgraphs by Guoli Ding
in [Subgraphs and well-quasi-ordering, Journal of Graph Theory, 16(5):489--502,
1992] and for induced subgraphs by Peter Damaschke in [Induced subgraphs and
well-quasi-ordering, Journal of Graph Theory, 14(4):427--435, 1990]
Induced minors and well-quasi-ordering
International audienceA graph H is an induced minor of a graph G if it can be obtained from an induced subgraph of G by contracting edges. Otherwise, G is said to be H-induced minor-free. Robin Thomas showed in [Graphs without K 4 and well-quasi-ordering, Journal of Combinatorial Theory, Series B, 38(3):240 – 247, 1985] that K 4-induced minor-free graphs are well-quasi ordered by induced minors. We provide a dichotomy theorem for H-induced minor-free graphs and show that the class of H-induced minor-free graphs is well-quasi-ordered by the induced minor relation if and only if H is an induced minor of the gem (the path on 4 vertices plus a dominating vertex) or of the graph obtained by adding a vertex of degree 2 to the complete graph on 4 vertices.Similar dichotomy results were previously given by Guoli Ding in [Subgraphs and well-quasi-ordering, Journal of Graph Theory, 16(5):489–502, 1992] for subgraphs and Peter Damaschke in [Induced subgraphs and well-quasi-ordering, Journal of Graph Theory, 14(4):427–435, 1990] for induced subgraphs
Tame Berge trigraphes
L'objectif de cette thèse est de réussir à utiliser des décompositions de graphes afin de résoudre des problèmes algorithmiques sur les graphes. Notre objet d'étude principal est la classe des graphes de Berge apprivoisés. Les graphes de Berge sont les graphes ne possédant ni cycle de longueur impaire supérieur à 4 ni complémentaire de cycle de longueur impaire supérieure à 4. Dans les années 60, Claude Berge a conjecturé que les graphes de Berge étaient des graphes parfaits. C'est-à-dire que la taille de la plus grande clique est exactement le nombre minimum de couleurs nécessaire à une coloration propre et ce pour tout sous-graphe. En 2002, Chudnovsky, Robertson, Seymour et Thomas ont démontré cette conjecture en utilisant un théorème de structure: les graphes de Berge sont basiques ou admettent une décomposition. Ce résultat est très utile pour faire des preuves par induction. Cependant, une des décompositions du théorème, la skew-partition équilibrée, est très difficile à utiliser algorithmiquement. Nous nous focalisons donc sur les graphes de Berge apprivoisés, c'est-à-dire les graphes de Berge sans skew-partition équilibrée. Pour pouvoir faire des inductions, nous devons adapter le théorème destructure de Chudnovsky et al à notre classe. Nous prouvons un résultat plus fort: les graphes de Berge apprivoisés sont basiques ou admettent une décomposition telle qu'un côté de la décomposition soit toujours basique. Nous avons de plus un algorithme calculant cette décomposition. Nous utilisons ensuite notre théorème pour montrer que les graphes de Berge apprivoisés admettent la propriété du grand biparti, de la clique-stable séparation et qu'il existe un algorithme polynomial permettant de calculer le stable maximum.The goal of this these is to use graph's decompositions to solve algorithmic problems on graphs. We will study the class of Berge tame graphs. A Berge graph is a graph without cycle of odd length at least 4 nor complement of cycle of odd length at least 4.In the 60's, Claude Berge conjectured that Berge graphs are perfect graphs. The size of the biggest clique is exactly the number of colors required to color the graph. In 2002, Chudnovsky, Robertson, Seymour et Thomas proved this conjecture using a theorem of decomposition: Berge graphs are either basic or have a decomposition. This is a useful result to do proof by induction. Unfortunately, one of the decomposition, the skew-partition, is really hard to use. We arefocusing here on Berge tame graphs, i.e~Berge graph without balanced skew-partition. To be able to do induction, we must first adapt the Chudnovsky et al's theorem of structure to our class. We prove a stronger result: Berge tame graphs are basic or have a decomposition such that one side is always basic. We also have an algorithm to compute this decomposition. We then use our theorem to prouve that Berge tame graphs have the big-bipartite property, the clique-stable set separation property and there exists a polytime algorithm to compute the maximum stable set
Trigraphes de Berge apprivoisés
The goal of this these is to use graph's decompositions to solve algorithmic problems on graphs. We will study the class of Berge tame graphs. A Berge graph is a graph without cycle of odd length at least 4 nor complement of cycle of odd length at least 4.In the 60's, Claude Berge conjectured that Berge graphs are perfect graphs. The size of the biggest clique is exactly the number of colors required to color the graph. In 2002, Chudnovsky, Robertson, Seymour et Thomas proved this conjecture using a theorem of decomposition: Berge graphs are either basic or have a decomposition. This is a useful result to do proof by induction. Unfortunately, one of the decomposition, the skew-partition, is really hard to use. We arefocusing here on Berge tame graphs, i.e~Berge graph without balanced skew-partition. To be able to do induction, we must first adapt the Chudnovsky et al's theorem of structure to our class. We prove a stronger result: Berge tame graphs are basic or have a decomposition such that one side is always basic. We also have an algorithm to compute this decomposition. We then use our theorem to prouve that Berge tame graphs have the big-bipartite property, the clique-stable set separation property and there exists a polytime algorithm to compute the maximum stable set.L'objectif de cette thèse est de réussir à utiliser des décompositions de graphes afin de résoudre des problèmes algorithmiques sur les graphes. Notre objet d'étude principal est la classe des graphes de Berge apprivoisés. Les graphes de Berge sont les graphes ne possédant ni cycle de longueur impaire supérieur à 4 ni complémentaire de cycle de longueur impaire supérieure à 4. Dans les années 60, Claude Berge a conjecturé que les graphes de Berge étaient des graphes parfaits. C'est-à-dire que la taille de la plus grande clique est exactement le nombre minimum de couleurs nécessaire à une coloration propre et ce pour tout sous-graphe. En 2002, Chudnovsky, Robertson, Seymour et Thomas ont démontré cette conjecture en utilisant un théorème de structure: les graphes de Berge sont basiques ou admettent une décomposition. Ce résultat est très utile pour faire des preuves par induction. Cependant, une des décompositions du théorème, la skew-partition équilibrée, est très difficile à utiliser algorithmiquement. Nous nous focalisons donc sur les graphes de Berge apprivoisés, c'est-à-dire les graphes de Berge sans skew-partition équilibrée. Pour pouvoir faire des inductions, nous devons adapter le théorème destructure de Chudnovsky et al à notre classe. Nous prouvons un résultat plus fort: les graphes de Berge apprivoisés sont basiques ou admettent une décomposition telle qu'un côté de la décomposition soit toujours basique. Nous avons de plus un algorithme calculant cette décomposition. Nous utilisons ensuite notre théorème pour montrer que les graphes de Berge apprivoisés admettent la propriété du grand biparti, de la clique-stable séparation et qu'il existe un algorithme polynomial permettant de calculer le stable maximum
Well-quasi-ordering -contraction-free graphs
International audienceA well-quasi-order is an order which contains no infinite decreasing sequence and no infinite collection of incomparable elements. In this paper, we consider graph classes defined by excluding one graph as contraction. More precisely, we give a complete characterization of graphs H such that the class of H-contraction-free graphs is well-quasi-ordered by the contraction relation. This result is the contraction analogue on the previous dichotomy theorems of Damsaschke [Induced subgraphs and well-quasi-ordering, Journal of Graph Theory, 14(4):427-435, 1990] on the induced subgraph relation, Ding [Subgraphs and well-quasi-ordering, Journal of Graph Theory, 16(5):489-502, 1992] on the subgraph relation, and B{\l}asiok et al. [Induced minors and well-quasi-ordering, ArXiv e-prints, 1510.07135, 2015] on the induced minor relation
Coloring perfect graphs with no balanced skew-partitions
International audienceWe present an algorithm that computes a maximum stable set of any perfect graph with no balanced skew-partition. We present time algorithm that colors them