Simplicial cliques in claw-free graphs

A graph G is claw-free if no induced subgraph of it is isomorphic to the complete bipartite graph K1,3, and it is prime if |V (G)| ≥ 4 and there is no X ⊆ V (G) with 1 < |X| < |V (G)| such that every vertex of V (G) \X with a neighbour in X is adjacent to every x ∈ X. A simplicial clique in G is a non-empty clique K, such that for every k ∈ K the set of neighbours of k in V (G) \K is a clique. We prove that a prime claw-free graph G has at most |V (G)|+ 1 simplicial cliques, and give an algorithm to find them all with running time O(|V (G)|4). We also prove a lemma, that we hope is of independent interest, that if G is a prime graph that is not a member of a particular family of exceptions, and H is a prime induced subgraph of G, then (up to isomorphism) G can be grown from H, adding one vertex at a time, in such a way that all the graphs constructed along the way are prime induced subgraphs of G. Finally, we apply our results to claw-free graphs that are not prime. Such a graph may have exponentially many simplicial cliques, so in polynomial time we cannot list them all, but we can in a sense describe them all.