Complexity of injective homomorphisms to small tournaments, and of injective oriented colourings

TL;DR

This paper investigates the computational complexity of locally-injective homomorphisms and colourings from oriented graphs to small fixed tournaments, establishing a dichotomy theorem for each definition of local injectivity.

Contribution

It provides a comprehensive complexity classification for various local injectivity definitions in oriented graph homomorphisms to small tournaments, including dichotomy theorems.

Findings

Decidability of existence varies with the definition of local injectivity.

Dichotomy theorems are established for each local injectivity case.

Complexity classifications are provided for homomorphisms to tournaments with up to three vertices.

Abstract

Several possible definitions of local injectivity for a homomorphism of an oriented graph $G$ to an oriented graph $H$ are considered. In each case, we determine the complexity of deciding whether there exists such a homomorphism when $G$ is given and $H$ is a fixed tournament on three or fewer vertices. Each possible definition leads to a locally-injective oriented colouring problem. A dichotomy theorem is proved in each case.