The complexity of locally injective homomorphisms

A homomorphism f:G→H, from a digraph G to a digraph H, is locally injective if the restriction of f to N⁻(v) is an injective mapping, for each v∈V(G). The problem of deciding whether such an f exists is known as the injective H-colouring problem (INJ-HOM_H). In this paper, we classify the problem INJ-HOM_H as being either a problem in P or a problem that is NP-complete. This is done in the case where H is a reflexive digraph (i.e. H has a loop at every vertex) and in the case where H is an irreflexive tournament. A full classification in the irreflexive case seems hard, and we provide some evidence as to why this may be the case.