. An independent set of three vertices such that each pair is joined by a path that avoids the neighborhood of the third is called an asteroidal triple. A graph is asteroidal triple-free (AT-free, for short) if it contains no asteroidal triples. The motivation for this investigation was provided, in part, by the fact that the asteroidal triple-free graphs provide a common generalization of interval, permutation, trapezoid, and cocomparability graphs. The main contribution of this work is to investigate and reveal fundamental structural properties of AT-free graphs. Specifically, we show that every connected AT-free graph contains a dominating pair, that is, a pair of vertices such that every path joining them is a dominating set in the graph. We then provide characterizations of AT-free graphs in terms of dominating pairs and minimal triangulations. Subsequently, we state and prove a decomposition theorem for AT-free graphs. An assortment of other properties of AT-free graphs is also p...

An asteroidal triple (AT) is a set of three vertices such that there is a path between any pair of them avoiding the closed neighborhood of the third. A graph is called AT-free if it does not have an AT. We show that there is an<F3.502e+05><F3.817e+05><F3.502e+05> O(n<F2.756e+05> 4<F3.817e+05> ) time algorithm to compute the maximum weight of an independent set for AT-free graphs. Furthermore, we obtain<F3.502e+05><F3.817e+05><F3.502e+05> O(n<F2.756e+05> 4<F3.817e+05> ) time algorithms to solve the<F3.728e+05> independent dominating set<F3.817e+05> and the<F3.728e+05> independent perfect dominating set<F3.817e+05> problems on AT-free graphs. We also show how to adapt these algorithms such that they solve the corresponding problem for graphs with bounded asteroidal number in polynomial time. Finally, we observe that the problems clique and partition into cliques remain NP-complete when restricted to AT-free graphs.