Geodesic
Simplicial and nonsimplicial complete subgraphs
Discussiones Mathematicae. Graph Theory, Tome 31 (2011) no. 3, pp. 577-586.

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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.
Keywords: simplicial clique, strongly chordal graph, trivially perfect graph, hereditary clique-Helly graph, strong p-Helly graph 
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McKee, Terry. Simplicial and nonsimplicial complete subgraphs. Discussiones Mathematicae. Graph Theory, Tome 31 (2011) no. 3, pp. 577-586. http://geodesic.mathdoc.fr/item/DMGT_2011_31_3_a11/

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