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Discussiones Mathematicae Graph Theory 31(3) (2011) 577-586
doi: 10.7151/dmgt.1566

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Simplicial and Nonsimplicial Complete Subgraphs

Terry A. McKee

Department of Mathematics & Statistics
Wright State University
Dayton, Ohio 45435, USA


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.

Keywords: simplicial clique, strongly chordal graph, trivially perfect graph, hereditary clique-Helly graph, strong p-Helly graph

2010 Mathematics Subject Classification: 05C75 (05C69).


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Received 30 March 2010
Accepted 2 September 2010