Asteroidal Sets and Dominating Paths

Oleksiy Al-saadi, Jamie Radcliffe

Abstract: An independent set of three vertices is called an asteroidal triple (AT) if there exists a path between any two of them that avoids the neighborhood of the third. Asteroidal triple-free (AT-free) graphs are very well-studied, but some of their various superclasses are not. We study two of these superclasses: hereditary dominating pair (HDP) graphs and diametral path graphs. We correct a mistake that has appeared in the literature claiming that the class of diametral path graphs are a superclass of HDP. More specifically, we show that a graph with a dominating shortest path does not necessarily contain a dominating diametral path. We say a graph is a strict dominating pair graph if it contains a dominating pair but has no dominating diametral path, and we show structural and algorithmic properties of these graphs. To study properties of HDP graphs, we introduce the notion of spread in asteroidal triples. Given a dominating pair, we show that all paths between this pair meet the common neighborhood of some pair from each asteroidal triple. We use these results to improve the best known time complexity for the recognition of chordal HDP graphs.