  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.
