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.