Locally Constrained Homomorphisms on Graphs of Bounded Treewidth and   Bounded Degree
  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. 
Steven Chaplick, Ji\v{r}\'i Fiala, Pim van 't Hof, Dani\"el Paulusma,   Marek Tesa\v{r}
cs.CC cs.DM math.CO