Powers of Asteroidal Triple-free Graphs with Applications

Abstract

An asteroidal triple is an independent set of three vertices in a graph such that every two of them are joined by a path avoiding the closed neighborhood of the third. Graphs without asteroidal triples are called AT-free graphs. In this paper, we show that every AT-free graph admits a vertex ordering that we call a $2$-cocomparability ordering. The new suggested ordering generalizes the cocomparability ordering achievable for cocomparability graphs. According to the property of this ordering, we show that every proper power $G^k$ ($k\ge 2$) of an AT-free graph $G$ is a cocomparability graph. Moreover, we demonstrate that our results can be exploited for algorithmic purposes on AT-free graphs.