Partial characterizations of 1-perfectly orientable graphs

TL;DR

This paper explores the structure of 1-perfectly orientable graphs, providing characterizations, identifying preserving transformations, and describing their behavior within specific graph classes, advancing understanding of this polynomial-time recognizable class.

Contribution

It offers new structural characterizations, identifies transformations that preserve the class, and characterizes 1-perfectly orientable graphs within cographs and cobipartite graphs.

Findings

Characterization via edge clique covers

Infinite family of minimal forbidden induced minors

Within cographs and cobipartite graphs, they coincide with known classes

Abstract

We study the class of 1-perfectly orientable graphs, that is, graphs having an orientation in which every out-neighborhood induces a tournament. 1-perfectly orientable graphs form a common generalization of chordal graphs and circular arc graphs. Even though they can be recognized in polynomial time, little is known about their structure. In this paper, we develop several results on 1-perfectly orientable graphs. In particular, we: (i) give a characterization of 1-perfectly orientable graphs in terms of edge clique covers, (ii) identify several graph transformations preserving the class of 1-perfectly orientable graphs, (iii) exhibit an infinite family of minimal forbidden induced minors for the class of 1-perfectly orientable graphs, and (iv) characterize the class of 1-perfectly orientable graphs within the classes of cographs and of cobipartite graphs. The class of 1-perfectly orientable co-bipartite graphs coincides with the class of co-bipartite circular arc graphs.