Simplicial and nonsimplicial complete subgraphs

Define a complete subgraph Q to be simplicial in a graph G when Q is contained in exactly one maximal complete subgraph (‘maxclique’) of G; otherwise, Q is nonsimplicial. Several graph classes—including strong p-Helly graphs and strongly chordal graphs—are shown to have pairs of peculiarly related new characterizations: (i) for every k ≥ 2, a certain property holds for the complete subgraphs that are in k or more maxcliques of G, and (ii) in every induced subgraph H of G, that same property holds for the nonsimplicial complete subgraphs of H . One example: G is shown to be hereditary clique-Helly if and only if, for every k ≥ 2, every triangle whose edges are each in k or more maxcliques is itself in k or more maxcliques; equivalently, in every induced subgraph H of G, if each edge of a triangle is nonsimplicial in H , then the triangle itself is nonsimplicial in H .