Obstructions for local tournament orientation completions

TL;DR

This paper characterizes minimal obstructions in partially oriented graphs that prevent completion to local tournaments, extending known graph characterizations and aiding understanding of orientation problems.

Contribution

It provides a complete list of obstructions for completing partially oriented graphs to local tournaments, expanding the theory of orientation completion problems.

Findings

Complete list of obstructions for local tournament completion

Extension of Tucker's forbidden subgraph characterization

Characterization of obstructions for acyclic local tournaments in a companion paper

Abstract

The orientation completion problem for a class of oriented graphs asks whether a given partially oriented graph can be completed to an oriented graph in the class by orienting the unoriented edges of the partially oriented graph. Orientation completion problems have been studied recently for several classes of oriented graphs, yielding both polynomial time solutions as well as NP-completeness results. Local tournaments are a well-structured class of oriented graphs that generalize tournaments and their underlying graphs are intimately related to proper circular-arc graphs. According to Skrien, a connected graph can be oriented as a local tournament if and only if it is a proper circular-arc graph. Proper interval graphs are precisely the graphs which can be oriented as acyclic local tournaments. It has been proved that the orientation completion problems for the classes of local tournaments and acyclic local tournaments are both polynomial time solvable. In this paper we characterize the partially oriented graphs that can be completed to local tournaments by determining the complete list of obstructions. These are in a sense minimal partially oriented graphs that cannot be completed to local tournaments. The result may be viewed as an extension of the well-known forbidden subgraph characterization of proper circular-arc graphs obtained by Tucker. The complete list of obstructions for acyclic local tournament orientation completions has been given in a companion paper.