Vertex-transitive CIS graphs

TL;DR

This paper characterizes vertex-transitive CIS graphs, establishing their equivalence to being well-covered, co-well-covered, and satisfying a specific product condition, and classifies certain subclasses including irreducible and low-valency graphs.

Contribution

It provides a complete characterization of vertex-transitive CIS graphs and classifies irreducible well-covered CIS graphs with small clique number and low valency vertex-transitive CIS graphs.

Findings

Vertex-transitive CIS graphs are characterized by being well-covered, co-well-covered, and satisfying a product condition.

Classification of irreducible well-covered CIS graphs with clique number at most 3.

Identification of an infinite family of vertex-transitive CIS graphs that are not Cayley graphs.

Abstract

A CIS graph is a graph in which every maximal stable set and every maximal clique intersect. A graph is well-covered if all its maximal stable sets are of the same size, co-well-covered if its complement is well-covered, and vertex-transitive if, for every pair of vertices, there exists an automorphism of the graph mapping one to the other. We show that a vertex-transitive graph is CIS if and only if it is well-covered, co-well-covered, and the product of its clique and stability numbers equals its order. A graph is irreducible if no two distinct vertices have the same neighborhood. We classify irreducible well-covered CIS graphs with clique number at most 3 and vertex-transitive CIS graphs of valency at most 7, which include an infinite family. We also exhibit an infinite family of vertex-transitive CIS graphs which are not Cayley.