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A graph is CIS if every maximal clique interesects every maximal stable set.Currently, no good characterization or recognition algorithm for the CIS graphsis known. We characterize graphs in which every maximal matching saturates allvertices of degree at least two and use this result to give a structural,efficiently testable characterization of claw-free CIS graphs. We answer in thenegative a question of Dobson, Hujdurovi\'c, Milani\v{c}, and Verret[Vertex-transitive CIS graphs, European J. Combin. 44 (2015) 87-98] askingwhether the number of vertices of every CIS graph is bounded from above by theproduct of its clique and stability numbers. On the positive side, we show thatthe question of Dobson et al. has an affirmative answer in the case ofclaw-free graphs.

A characterization of claw-free CIS graphs and new results on the order of CIS graphs