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Complexity of Injective Homomorphisms to Small Tournaments, and of Injective Oriented Colourings

DOI: 10.4236/ojdm.2023.131001, PP. 1-15

Keywords: Injective Graph Homomorphism, Oriented Colouring, Complexity

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Abstract:

Several possible definitions of local injectivity for a homomorphism of an oriented graph G to an oriented graph H are considered. In each case, we determine the complexity of deciding whether there exists such a homomorphism when G is given and H is a fixed tournament on three or fewer vertices. Each possible definition leads to a locally-injective oriented colouring problem. A dichotomy theorem is proved in each case.

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