On well-covered direct products

A graph $G$ is well-covered if all maximal independent sets of $G$ have the same cardinality. In 1992 Topp and Volkmann investigated the structure of well-covered graphs that have nontrivial factorizations with respect to some of the standard graph products. In particular, they showed that both factors of a well-covered direct product are also well-covered and proved that the direct product of two complete graphs (respectively, two cycles) is well-covered precisely when they have the same order (respectively, both have order 3 or 4). Furthermore, they proved that the direct product of two well-covered graphs with independence number one-half their order is well-covered. We initiate a characterization of nontrivial, connected well-covered graphs $G$ and $H$, whose independence numbers are strictly less than one-half their orders, such that their direct product $G \times H$ is well-covered. In particular, we show that in this case both $G$ and $H$ have girth 3 and we present several infinite families of such well-covered direct products. Moreover, we show that if $G$ is a factor of any well-covered direct product, then $G$ is a complete graph unless it is possible to create an isolated vertex by removing the closed neighborhood of some independent set of vertices in $G$.