Conflict-free graph orientations with parity constraints

TL;DR

This paper investigates the problem of assigning orientations to multigraphs with parity constraints and additional conflict restrictions, providing complexity results and efficient algorithms for certain cases.

Contribution

It introduces the concept of conflict constraints in graph orientations and offers algorithms for cases with disjoint conflict pairs, advancing understanding of parity-constrained orientations.

Findings

Deciding parity orientations with conflicts is NP-complete for conflict sets of size two or more.

Efficient algorithms are developed for orientations with disjoint conflict pairs.

The study extends known results on even orientations to more complex constrained scenarios.

Abstract

It is known that every multigraph with an even number of edges has an even orientation (i.e., all indegrees are even). We study parity constrained graph orientations under additional constraints. We consider two types of constraints for a multigraph G=(V,E): (1) an exact conflict constraint is an edge set C in E and a vertex v in V such that C should not equal the set of incoming edges at v; (2) a subset conflict constraint is an edge set C in E and a vertex v in V such that C should not be a subset of incoming edges at v. We show that it is NP-complete to decide whether G has an even orientation with exact or subset conflicts, for all conflict sets of size two or higher. We present efficient algorithms for computing parity constrained orientations with disjoint exact or subset conflict pairs.