On dominating set polyhedra of circular interval graphs

TL;DR

This paper provides a complete linear description of the dominating set polytope for circular interval graphs, extending previous work on stable set polytopes and covering polyhedra for this class of graphs.

Contribution

It introduces a comprehensive linear description of the dominating set polytope on circular interval graphs, generalizing to larger classes of covering polyhedra with circular matrices.

Findings

Complete linear description of the dominating set polytope obtained.

Results extend to various dominating set problem variants.

Connections established between stable set and dominating set polytopes.

Abstract

Clique-node and closed neighborhood matrices of circular interval graphs are circular matrices. The stable set polytope and the dominating set polytope on these graphs are therefore closely related to the set packing polytope and the set covering polyhedron on circular matrices. Eisenbrand et al. take advantage of this relationship to propose a complete linear description of the stable set polytope on circular interval graphs. In this paper we follow similar ideas to obtain a complete description of the dominating set polytope on the same class of graphs. As in the packing case, our results are established for a larger class of covering polyhedra of the form Q*(A,b):= conv{x \in \Z^n_+: Ax >= b}, with A a circular matrix and b an integer vector. These results also provide linear descriptions of polyhedra associated with several variantsof the dominating set problem on circular interval graphs.