Asteroidal triples of moplexes

An asteroidal triple is an independent set of vertices such that each pair is joined by a path that avoids the neighborhood of the third, and a moplex is an extension to an arbitrary graph of a simplicial vertex in a triangulated graph. The main result of this paper is that the investigation of the set of moplexes of a graph is sufficient to conclude as to its having an asteroidal triple. Specifically, we show that a graph has an asteroidal triple of vertices if and only if it has an asteroidal triple of moplexes. We also examine the behavior of an asteroidal triple of moplexes in the course of a minimal triangulation process, and give some related properties.