  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.
