20 research outputs found
Decomposing Berge graphs
URL des Cahiers : https://halshs.archives-ouvertes.fr/CAHIERS-MSECahiers de la Maison des Sciences Economiques 2006.02 - ISSN 1624-0340A hole in a graph is an induced cycle on at least four vertices. A graph is Berge if it has no old hole and if its complement has no odd hole. In 2002, Chudnovsky, Robertson, Seymour and Thomas proved a decomposition theorem for Berge graphs saying that every Berge graph either is in a well understood basic class or has some kind of decomposition. Then, Chudnovsky proved a stronger theorem by restricting the allowed decompositions and another theorem where some decompositions were restricted while other decompositions were extended. We prove here a theorem stronger than all those previously known results. Our proof uses at an essential step one of the theorems of Chudnovsky.Un trou dans un graphe est un cycle induit avec au moins quatre sommets. Un graphe est de Berge si ni lui ni son complémentaire ne contiennent de trou impair. En 2002, Chudnovsky, Robertson, Seymour et Thomas ont prouvé un théorème de décomposition affirmant que tout graphe de Berge, ou bien appartient à une classe basique bien comprise, ou bien admet un certain type de décomposition. Puis Chudnovsky a prouvé un théorème plus fort en restreignant les décompositions autorisées. Chudnovsky a aussi prouvé un autre théorème où certaines décompositions sont restreintes tandis que d'autres sont étendues. Nous prouvons ici un théorème plus fort que ces trois résultats, en ce sens qu'il les implique facilement. Notre preuve utilise l'un des théorèmes de Chudnovsky
The Erd\H{o}s-Hajnal Conjecture for Paths and Antipaths
We prove that for every k, there exists such that every graph G on n
vertices not inducing a path and its complement contains a clique or a
stable set of size
Even pairs in Berge graphs
Abstract Our proof (with Robertson and Thomas) of the strong perfect graph conjecture ran to 179 pages of dense matter; and the most impenetrable part was the final 55 pages, on what we called "wheel systems". In this paper we give a replacement for those 55 pages, much easier and shorter, using "even pairs". This is based on an approach of Maffray and Trotignon
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