Asteroidal Triple-Free Graphs

Many families of perfect graphs such as interval graphs, permutation graphs, trapezoid graphs and cocomparability graphs demonstrate a type of linear ordering of their vertex sets. These graphs are all subfamilies of a class of graphs called the asteroidal triple-free graphs. (An independent triple {x, y, z} is called an asteroidal triple (AT, for short) if between any pair in the triple there exists a path that avoids the neighbourhood of the third vertex.) In this paper we argue that the property of being AT-free is what is enforcing the linear ordering of the vertex sets. To justify this claim, we present various structural properties and characterizations of AT-free graphs.

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