On CIS Circulants

TL;DR

This paper investigates the structure of CIS circulant graphs, providing new constructions of such graphs and exploring their properties, including their closure under certain graph operations and their relation to well-covered graphs.

Contribution

The paper introduces a non-trivial infinite family of CIS circulants and analyzes their structure, expanding understanding of CIS graphs within circulants.

Findings

All P_4-free circulants are CIS.

Constructed an infinite family of sparse CIS circulants.

CIS circulants are closed under complement and lexicographic product.

Abstract

A circulant is a Cayley graph over a cyclic group. A well-covered graph is a graph in which all maximal stable sets are of the same size, or in other words, they are all maximum. A CIS graph is a graph in which every maximal stable set and every maximal clique intersect. It is not difficult to show that a circulant G is a CIS graph if and only if G and its complement are both well-covered and the product of the independence and the clique numbers of G is equal to the number of vertices. It is also easy to demonstrate that both families, the circulants and the CIS graphs, are closed with respect to the operations of taking the complement and lexicographic product. We study the structure of the CIS circulants. It is well-known that all P_4-free graphs are CIS. In this paper, in addition to the simple family of the P_4-free circulants, we construct a non-trivial sparse but infinite family of CIS circulants. We are not aware of any CIS circulant that could not be obtained from graphs in this family by the operations of taking the complement and lexicographic product.