On the powers of graphs with bounded asteroidal number Original Research Article

For an undirected graph G=(V,E), the kth power Gk is the graph with the same vertex set as G such that two vertices are adjacent in Gk 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−NG[a], where NG[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 Gk∈A(s) for s⩾2, then so is Gk+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.