Abstract
An asteroidal triple (AT) is a set of vertices such that each pair of vertices is joined by a path that avoids the neighborhood of the third. Every AT-free graph contains a dominating pair, a pair of vertices such that for every path between them, every vertex of the graph is within distance one of the path. We say that a graph is a hereditary dominating pair (HDP) graph if each of its connected induced subgraphs contains a dominating pair. In this paper we introduce the notion of frame HDP graphs in order to capture the structure of HDP graphs that contain asteroidal triples. We also determine the maximum diameter of frame HDP graphs.
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