1-perfectly orientable graphs and graph products

TL;DR

This paper characterizes when standard graph products (Cartesian, strong, direct, lexicographic) of two graphs are 1-perfectly orientable, a class generalizing chordal and circular arc graphs, despite the lack of structural characterization.

Contribution

It provides the first structural characterizations of 1-perfectly orientable graphs resulting from standard graph products.

Findings

Characterization for Cartesian product being 1-p.o.

Characterization for strong product being 1-p.o.

Characterization for direct and lexicographic products being 1-p.o.

Abstract

A graph G is said to be 1-perfectly orientable (1-p.o. for short) if it admits an orientation such that the out-neighborhood of every vertex is a clique in G. The class of 1-p.o. graphs forms a common generalization of the classes of chordal and circular arc graphs. Even though 1-p.o. graphs can be recognized in polynomial time, no structural characterization of 1-p.o. graphs is known. In this paper we consider the four standard graph products: the Cartesian product, the strong product, the direct product, and the lexicographic product. For each of them, we characterize when a nontrivial product of two graphs is 1-p.o.