356 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
Complexity of colouring problems restricted to unichord-free and \{square,unichord\}-free graphs
A \emph{unichord} in a graph is an edge that is the unique chord of a cycle.
A \emph{square} is an induced cycle on four vertices. A graph is
\emph{unichord-free} if none of its edges is a unichord. We give a slight
restatement of a known structure theorem for unichord-free graphs and use it to
show that, with the only exception of the complete graph , every
square-free, unichord-free graph of maximum degree~3 can be total-coloured with
four colours. Our proof can be turned into a polynomial time algorithm that
actually outputs the colouring. This settles the class of square-free,
unichord-free graphs as a class for which edge-colouring is NP-complete but
total-colouring is polynomial
Strong arc decompositions of split digraphs
A {\bf strong arc decomposition} of a digraph is a partition of its
arc set into two sets such that the digraph is
strong for . Bang-Jensen and Yeo (2004) conjectured that there is some
such that every -arc-strong digraph has a strong arc decomposition. They
also proved that with one exception on 4 vertices every 2-arc-strong
semicomplete digraph has a strong arc decomposition. Bang-Jensen and Huang
(2010) extended this result to locally semicomplete digraphs by proving that
every 2-arc-strong locally semicomplete digraph which is not the square of an
even cycle has a strong arc decomposition. This implies that every 3-arc-strong
locally semicomplete digraph has a strong arc decomposition. A {\bf split
digraph} is a digraph whose underlying undirected graph is a split graph,
meaning that its vertices can be partioned into a clique and an independent
set. Equivalently, a split digraph is any digraph which can be obtained from a
semicomplete digraph by adding a new set of vertices and some
arcs between and . In this paper we prove that every 3-arc-strong split
digraph has a strong arc decomposition which can be found in polynomial time
and we provide infinite classes of 2-strong split digraphs with no strong arc
decomposition. We also pose a number of open problems on split digraphs
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