Complexity of locally-injective homomorphisms to tournaments

TL;DR

This paper investigates the computational complexity of locally-injective homomorphisms from oriented graphs to reflexive tournaments, establishing NP-completeness for larger tournaments and polynomial solvability for smaller ones.

Contribution

It characterizes the complexity of locally-injective homomorphism problems to reflexive tournaments, showing NP-completeness for tournaments with three or more vertices and polynomial-time solvability for smaller ones.

Findings

NP-complete for reflexive tournaments with ≥3 vertices

Polynomial-time solvable for reflexive tournaments with ≤2 vertices

Analyzes two definitions of local-injectivity

Abstract

For oriented graphs $G$ and $H$, a homomorphism $f: G \rightarrow H$ is locally-injective if, for every $v \in V(G)$, it is injective when restricted to some combination of the in-neighbourhood and out-neighbourhood of $v$. Two of the possible definitions of local-injectivity are examined. In each case it is shown that the associated homomorphism problem is NP-complete when $H$ is a reflexive tournament on three or more vertices with a loop at every vertex, and solvable in polynomial time when $H$ is a reflexive tournament on two or fewer vertices.