Orientations without forbidden patterns on three vertices
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Given a set of oriented graphs F, a graph G is an F-graph if it admits an F-free orientation. Building on previous work by Bang-Jensen and Urrutia, we propose a master algorithm that determines if a graph admits an F-free orientation when F is a subset of the orientations of P_3 and the transitive triangle. We extend previous results of Skrien by studying the class of F-graphs, when F is any set of oriented graphs of order three; structural characterizations for all such sets are provided, except for the so-called perfectly-orientable graphs and one of its subclasses, which remain as open problems.
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