Obstructions to some injective oriented colourings

TL;DR

This paper investigates various definitions of local injectivity in oriented graph homomorphisms, identifying obstructions and characterizing when injective oriented colourings are possible in polynomial time.

Contribution

It characterizes obstructions for polynomial-time solvable injective oriented colouring problems using forbidden oriented subgraphs.

Findings

Identifies sets of oriented graphs characterizing injective colourability

Provides polynomial-time solvability conditions for certain injective colouring problems

Establishes a framework for understanding obstructions in oriented graph colourings

Abstract

Each of several possible definitions of local injectivity for a homomorphism of an oriented graph $G$ to an oriented graph $H$ leads to an injective oriented colouring problem. For each case in which such a problem is solvable in polynomial time, we identify a set $\mathcal{F}$ of oriented graphs such that an oriented graph $G$ has an injective oriented colouring with the given number of colours if and only if there is no $F \in \mathcal{F}$ for which there is a locally-injective homomorphism of $F$ to $G$.