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.

💡 Research Summary

This paper investigates three variants of graph homomorphisms that impose local constraints on the mapping: locally bijective (LBHom), locally surjective (LSHom), and locally injective (LIHom). A homomorphism φ : G → H is locally bijective if, for every vertex u of G, the restriction of φ to the neighbourhood N_G(u) is a bijection onto N_H(φ(u)). The locally surjective and locally injective notions are defined analogously, requiring the neighbourhood mapping to be surjective or injective, respectively. These concepts arise in several areas such as distributed computing, network design, and social science, and they generalize ordinary graph homomorphisms by adding a stringent local structure.

The authors first address the computational complexity of deciding whether a given pair (G, H) admits such a locally constrained homomorphism. Using a reduction from the strongly NP‑complete 3‑Partition problem, they construct, for each variant, a pair of graphs with very restricted structural parameters. In the LBHom reduction, G has pathwidth at most 5 and H has pathwidth at most 3; in the LSHom reduction, G’s pathwidth is bounded by 4; and for LIHom, both G and H have pathwidth at most 2. The construction encodes each integer a_i of the 3‑Partition instance as a cycle of length b together with auxiliary vertices that enforce degree preservation. A central vertex x (and auxiliary vertices y, z) in G is linked to these structures so that any locally bijective mapping must map the set of vertices at distance two from x onto exactly one of the m cycles in H, thereby realizing a partition of the integers into m triples of sum b. The converse direction builds a homomorphism from a valid partition. Because the reduction respects degrees, the same gadgets also yield NP‑completeness when both G and H have maximum degree 3, showing that degree bounds alone do not simplify the problem.

These hardness results demonstrate that even on graphs that are “almost trees” (small pathwidth) or on subcubic graphs, the three locally constrained homomorphism problems remain computationally intractable.

On the positive side, the paper shows that when the guest graph G has bounded treewidth and either G or the host graph H has bounded maximum degree, all three problems become polynomial‑time solvable. The algorithmic framework relies on a nice tree decomposition of G. For each bag of the decomposition, the algorithm stores a compact description of the partial mapping of vertices in the bag to colors (vertices of H) together with the induced local constraints. Because the degree bound limits the size of each neighbourhood, the number of possible local configurations is a function only of the treewidth and the degree bound, yielding a dynamic‑programming table of polynomial size. The authors give explicit recurrence relations for introduce, forget, and join nodes, handling the bijectivity, surjectivity, or injectivity requirements locally at each step. Consequently, the running time is O(f(tw(G),Δ)·|V(G)|), where f is exponential only in the combined parameter (treewidth, maximum degree) but independent of the overall graph size.

A particularly noteworthy corollary is that LIHom can be solved in polynomial time when G is a tree (treewidth 1), which matches the known tractability of LBHom and LSHom on trees but contrasts with the NP‑completeness of LIHom on proper interval graphs. This result underscores that the treewidth bound is tight for LIHom.

The paper concludes with several open questions. The complexity of LBHom and LSHom when the host graph H is fixed remains only partially classified; the authors suggest investigating other structural parameters such as clique‑width, graph minors, or parameterized complexity (FPT) beyond the combined treewidth‑degree setting. They also note the potential for extending the dynamic‑programming approach to broader classes of locally constrained mappings.

In summary, the work delivers a comprehensive picture: it establishes strong NP‑hardness for locally bijective, surjective, and injective homomorphisms even on highly restricted graph families, while simultaneously identifying a robust tractable regime defined by bounded treewidth together with a bounded degree condition on either side of the homomorphism. This dual contribution advances both the theoretical understanding of locally constrained homomorphisms and provides practical algorithms for cases where the underlying graphs possess limited structural complexity.