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.
@article{bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1566,
author = {Terry A. McKee},
title = {Simplicial and nonsimplicial complete subgraphs},
journal = {Discussiones Mathematicae Graph Theory},
volume = {31},
year = {2011},
pages = {577-586},
zbl = {1229.05237},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1566}
}
Terry A. McKee. Simplicial and nonsimplicial complete subgraphs. Discussiones Mathematicae Graph Theory, Tome 31 (2011) pp. 577-586. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1566/
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