Locally constrained homomorphisms on graphs of bounded treewidth and bounded degreeSteven Chaplick, Jiří Fiala, Pim van 't Hof, Daniël Paulusma and Marek Tesař
A preliminary version of this paper appeared in the proceedings of FCT 2013, 19th International Symposium on Fundamentals of Computation Theory (held on August 19-21, 2013 in Liverpool, United Kingdom), Lecture Notes in Computer Science, vol. 8070, pp. 121-132, 2013.
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 in the restricted case where 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.