An Algorithmic Framework for Locally Contrainted Homomorphisms

A homomorphism ϕ from a guest graph G to a host graph H is locally bijective, injective, or surjective if for every u ∈ V (G), the restriction of ϕ to the neighbourhood of u is bijective, injective, or surjective, respectively. We prove a number of new FPT (fixed-parameter tractable), W[1]-hard, and paraNP-complete results for the corresponding decision problems LBHom, LIHom, and LSHom by considering a hierarchy of parameters of the guest graph G. In this way we strengthen several existing results. For our FPT results, we develop a new algorithmic framework that involves a general ILP (integer linear program) model. We also use our framework to prove FPT results for the Role Assignment problem, which originates from social network theory and is closely related to locally surjective homomorphisms.