How Difficult Is It to Recognize CIS Graphs?

A graph $G$ is called $CIS$ if each maximal clique intersects each maximal stable set of $G$, with maximality taken with respect to set inclusion. CIS graphs resemble perfect graphs in several respects and have interesting applications in game theory. The complexity of recognizing CIS graphs was posed as an open problem by Chv'atal in the 1990s and has since led to conflicting conjectures. We settle the problem by showing that recognizing CIS graphs is $\mathsf{c}\mathsf{o}\mathsf{N}\mathsf{P}\text{-complete}$.