Geodesic
Complexity of locally-injective homomorphisms to tournaments
Discrete mathematics & theoretical computer science, Tome 20 (2018) no. 2.

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For oriented graphs $G$ and $H$, a homomorphism $f: G \rightarrow H$ is locally-injective if, for every $v \in V(G)$, 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. 
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     author = {Bard, Stefan and Bellitto, Thomas and Duffy, Christopher and MacGillivray, Gary and Yang, Feiran},
     title = {Complexity of locally-injective homomorphisms to tournaments},
     journal = {Discrete mathematics & theoretical computer science},
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Bard, Stefan; Bellitto, Thomas; Duffy, Christopher; MacGillivray, Gary; Yang, Feiran. Complexity of locally-injective homomorphisms to tournaments. Discrete mathematics & theoretical computer science, Tome 20 (2018) no. 2. doi : 10.23638/DMTCS-20-2-4. http://geodesic.mathdoc.fr/articles/10.23638/DMTCS-20-2-4/

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