Orientations without forbidden patterns on three vertices
Santiago Guzmán‐Pro, César Hernández‐Cruz · Applied Mathematics and Computation · 2024
Given a set F of oriented graphs, a graph G is a Forbe(F)-graph if it admits an F-free orientation. Skrien showed that proper-circular arc graphs, nested interval graphs and comparability graphs, correspond to Forbe(F)-graph classes for some set F of orientations of P3. Building on these results, we exhibit the list of all Forbe(F)-graph classes when F is a set of oriented graphs on three vertices. Structural characterizations for these classes are provided, except for the so-called perfectly-orientable graphs and the transitive-perfectly-orientable graphs, which remain as open problems.