Locally Constrained Homomorphisms on Graphs of Bounded Treewidth and Bounded Degree

TL;DR

This paper investigates the computational complexity of locally constrained graph homomorphisms, proving NP-completeness in general but identifying polynomial cases when graphs have bounded treewidth and degree.

Contribution

It establishes NP-completeness for locally constrained homomorphism problems and identifies polynomial-time solvable cases based on graph treewidth and degree bounds.

Findings

NP-completeness for locally bijective, surjective, and injective homomorphisms on bounded pathwidth graphs.

Polynomial-time algorithms for these problems when graphs have bounded treewidth and degree.

Hardness persists even with maximum degree 3 graphs.

Abstract

A homomorphism from a graph G to a graph H is locally bijective, surjective, or injective if its restriction to the neighborhood of every vertex of G is bijective, surjective, or injective, respectively. We prove that the problems of testing whether a given graph G allows a homomorphism to a given graph H that is locally bijective, surjective, or injective, respectively, are NP-complete, even when G has pathwidth at most 5, 4, or 2, respectively, or when both G and H have maximum degree 3. We complement these hardness results by showing that the three problems are polynomial-time solvable if G has bounded treewidth and in addition G or H has bounded maximum degree.