An Algorithmic Framework for Locally Constrained 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. The corresponding decision problems, LBHom, LIHom and LSHom, are well studied both on general graphs and on special graph classes. We prove a number of new FPT, W[1]-hard and para-NP-complete results by considering a hierarchy of parameters of the guest graph G. For our FPT results, we do this through the development of a new algorithmic framework that involves a general ILP model. To illustrate the applicability of the new framework, we also use it to prove FPT results for the Role Assignment problem, which originates from social network theory and is closely related to locally surjective homomorphisms.