Characterization and recognition of some opposition and coalition graph classes

TL;DR

This paper characterizes opposition and coalition graphs within certain classes, providing polynomial-time recognition algorithms and insights into their structure, advancing understanding of these graph classes related to perfect graphs.

Contribution

It offers new characterizations for opposition and coalition graphs on restricted classes and introduces polynomial-time recognition algorithms.

Findings

Polynomial-time recognition algorithms for opposition and coalition graphs.

Characterizations of opposition and coalition graphs on specific graph classes.

A good characterization for generalized opposition graphs.

Abstract

A graph is an opposition graph, respectively, a coalition graph, if it admits an acyclic orientation which puts the two end-edges of every chordless 4-vertex path in opposition, respectively, in the same direction. Opposition and coalition graphs have been introduced and investigated in connection to perfect graphs. Recognizing and characterizing opposition and coalition graphs are long-standing open problems. This paper gives characterizations for opposition graphs and coalition graphs on some restricted graph classes. Implicit in our arguments are polynomial time recognition algorithms for these graphs. We also give a good characterization for the so-called generalized opposition graphs.