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Graph analysis reveals NP-completeness in homomorphisms to local tournaments with directed cycles, indicating complex proofs are essential.
In this paper we give a graph theoretic proof of the fact that deciding whether a homomorphism exists to a fixed local tournament with at least two directed cycles is NP-complete. One of the main reasons for the graph theoretic proof is that it showcases all of the techniques that have been built up over the years in the study of the digraph homomorphism problem.
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MacGillivray et al. (2011) studied this question.
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