An Algorithmic Framework for Locally Constrained Homomorphisms

Laurent Bulteau, Konrad K. Dabrowski, Noleen Köhler, Sebastian Ordyniak, Daniël Paulusma

Abstract: A homomorphism $$\phi $$ from a guest graph G to a host graph H is locally bijective, injective or surjective if for every $$u\in V(G)$$ , the restriction of $$\phi $$ 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 $$\textsf{FPT}$$ , $$\textsf{W}$$ [1]-hard and para- $$\textsf{NP}$$ -complete results by considering a hierarchy of parameters of the guest graph G. For our $$\textsf{FPT}$$ 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 $$\textsf{FPT}$$ results for the Role Assignment problem, which originates from social network theory and is closely related to locally surjective homomorphisms.