Characterization of Word-Representable Graphs in Terms of Semi-transitive Orientations

Abstract

Recall that, by Theorem 3.4.3, a graph is permutationally representable if and only if it is a comparability graph. The current chapter offers, based on [74, 75], a generalization of this theorem to the case of arbitrary word-representable graphs and so-called semi-transitive orientations (see Theorem 4.1.8). These acyclic orientations, being a generalization of transitive orientations related to partial orders, provide a powerful tool for studying word-representable graphs, in particular, allowing us to reformulate word-representability problems in pure graph-theoretical terms (via certain orientations of graphs).