Complexity of locally-injective homomorphisms to tournaments

For oriented graphs $G$ and $H$, a homomorphism $f: G \rightarrow H$ islocally-injective if, for every $v \in V(G)$, it is injective when restrictedto some combination of the in-neighbourhood and out-neighbourhood of $v$. Twoof the possible definitions of local-injectivity are examined. In each case itis shown that the associated homomorphism problem is NP-complete when $H$ is areflexive tournament on three or more vertices with a loop at every vertex, andsolvable in polynomial time when $H$ is a reflexive tournament on two or fewervertices.Comment: 22 pages, 16 figure