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Computing the clique number of [formula omitted]-perfect graphs in polynomial time

Authors
Journal
European Journal of Combinatorics
0195-6698
Publisher
Elsevier
Volume
35
Identifiers
DOI: 10.1016/j.ejc.2013.06.025
Disciplines
  • Computer Science
  • Mathematics

Abstract

Abstract A main result of combinatorial optimization is that the clique and chromatic numbers of a perfect graph are computable in polynomial time (Grötschel et al., 1981) [7]. This result relies on polyhedral characterizations of perfect graphs involving the stable set polytope of the graph, a linear relaxation defined by clique constraints, and a semi-definite relaxation, the Theta-body of the graph. A natural question is whether the algorithmic results for perfect graphs can be extended to graph classes with similar polyhedral properties. We consider a superclass of perfect graphs, the a-perfect graphs, whose stable set polytope is given by constraints associated with generalized cliques. We show that for such graphs the clique number can be computed in polynomial time as well. The result strongly relies upon Fulkerson’s antiblocking theory for polyhedra and Lovász’s Theta function.

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