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 $K\subseteq B$ and $S\subseteq W$. 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

A simple Havel-Hakimi type algorithm to realize graphical degree sequences of directed graphs

One of the simplest ways to decide whether a given finite sequence of positive integers can arise as the degree sequence of a simple graph is the greedy algorithm of Havel and Hakimi. This note extends their approach to directed graphs. It also studies cases of some simple forbidden edge-sets. Finally, it proves a result which is useful to design an MCMC algorithm to find random realizations of prescribed directed degree sequences.Comment: 11 pages, 1 figure submitted to "The Electronic Journal of Combinatorics

On Box-Perfect Graphs

Let $G=(V,E)$ be a graph and let $A_G$ be the clique-vertex incidence matrix of $G$. It is well known that $G$ is perfect iff the system $A_{_G}\mathbf x\le \mathbf 1$, $\mathbf x\ge\mathbf0$ is totally dual integral (TDI). In 1982, Cameron and Edmonds proposed to call $G$ box-perfect if the system $A_{_G}\mathbf x\le \mathbf 1$, $\mathbf x\ge\mathbf0$ is box-totally dual integral (box-TDI), and posed the problem of characterizing such graphs. In this paper we prove the Cameron-Edmonds conjecture on box-perfectness of parity graphs, and identify several other classes of box-perfect graphs. We also develop a general and powerful method for establishing box-perfectness