On the powers of graphs with bounded asteroidal number

Abstract For an undirected graph G =( V , E ), the kth power G k is the graph with the same vertex set as G such that two vertices are adjacent in G k if and only if their distance in G is at most k. A set of vertices A ⊆ V is an asteroidal set if for every vertex a ∈ A , the set A ⧹{ a } is contained in one connected component of G − N G [ a ], where N G [ a ] is the closed neighborhood of a in G. The asteroidal number of a graph G is the maximum cardinality of an asteroidal set in G. The class of graphs with asteroidal number at most s is denoted by A (s) . In this paper, we show that if G k ∈ A (s) for s ⩾2, then so is G k +1 . This generalizes a previous result for the family of AT-free graphs. Moreover, we consider the forbidden configurations for the powers of graphs with bounded asteroidal number. Based on these forbidden configurations, we show that every proper power of AT-free graphs is perfect.