On CIS circulants

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 α=α(G)α=α(G), 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 GG is a CIS graph if and only if GG and its complement G¯ are both well-covered and the product α(G)α(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 the lexicographic product. We study the structure of the CIS circulants. It is well-known that all P4P4-free graphs are CIS. In this paper, in addition to the simple family of P4P4-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 the lexicographic product.