Complexity Of Locally-Infective Homomorphisms To Tournaments

For oriented graphs G and H, a homomorphism f : G -> H is locally-injective if, for every v is an element of V (G), it is injective when restricted to some combination of the in-neighbourhood and out-neighbourhood of v. Two of the possible definitions of local-injectivity are examined. In each case it is shown that the associated homomorphism problem is NP-complete when H is a reflexive tournament on three or more vertices with a loop at every vertex, and solvable in polynomial time when H is a reflexive tournament on two or fewer vertices.