6 research outputs found
Coloring vertices of a graph or finding a Meyniel obstruction
A Meyniel obstruction is an odd cycle with at least five vertices and at most
one chord. A graph is Meyniel if and only if it has no Meyniel obstruction as
an induced subgraph. Here we give a O(n^2) algorithm that, for any graph, finds
either a clique and coloring of the same size or a Meyniel obstruction. We also
give a O(n^3) algorithm that, for any graph, finds either aneasily recognizable
strong stable set or a Meyniel obstruction
Finding a Strong Stable Set or a Meyniel Obstruction in any Graph
A strong stable set in a graph is a stable set that contains a vertex of every maximal clique of . A Meyniel obstruction is an odd circuit with at least five vertices and at most one chord. Given a graph and a vertex of , we give a polytime algorithm to find either a strong stable set containing or a Meyniel obstruction in . This can then be used to find in any graph, a clique and colouring of the same size or a Meyniel obstruction
Perfectly contractile graphs and quadratic toric rings
Perfect graphs form one of the distinguished classes of finite simple graphs.
In 2006, Chudnovsky, Robertson, Saymour and Thomas proved that a graph is
perfect if and only if it has no odd holes and no odd antiholes as induced
subgraphs, which was conjectured by Berge. We consider the class
of graphs that have no odd holes, no antiholes and no odd stretchers as induced
subgraphs. In particular, every graph belonging to is perfect.
Everett and Reed conjectured that a graph belongs to if and only
if it is perfectly contractile. In the present paper, we discuss graphs
belonging to from a viewpoint of commutative algebra. In fact,
we conjecture that a perfect graph belongs to if and only if
the toric ideal of the stable set polytope of is generated by quadratic
binomials. Especially, we show that this conjecture is true for Meyniel graphs,
perfectly orderable graphs, and clique separable graphs, which are perfectly
contractile graphs.Comment: 10 page
Perfect Graphs
This chapter is a survey on perfect graphs with an algorithmic flavor. Our emphasis is on important classes of perfect graphs for which there are fast and efficient recognition and optimization algorithms. The classes of graphs we discuss in this chapter are chordal, comparability, interval, perfectly orderable, weakly chordal, perfectly contractile, and chi-bound graphs. For each of these classes, when appropriate, we discuss the complexity of the recognition algorithm and algorithms for finding a minimum coloring, and a largest clique in the graph and its complement