A Linear Time Algorithm to Compute a Dominating Path in an AT-free Graph

An independent set fx; y; zg is called an asteroidal triple if between any pair in the triple there exists a path that avoids the neighborhood of the third. A graph is referred to as AT-free if it does not contain an asteroidal triple. We present a simple linear-time algorithm to compute a dominating path in a connected AT-free graph. Keywords. asteroidal triple-free graphs, domination, algorithms 1 Introduction A number of families of graphs including interval graphs [10], permutation graphs [6], trapezoid graphs [3, 5], and cocomparability graphs [8] feature a type of linear ordering of their vertex sets. It is precisely this linear ordering that is exploited in a search for efficient algorithms on these classes of graphs [2, 5, 7, 8, 9, 11, 12]. As it turns out, the classes mentioned above are all subfamilies of a class of graphs called the asteroidal triple-free graphs (AT-free graphs, for short). An independent triple fx; y; zg is called an asteroidal triple if between any p..