The restricted homomorphism problem RHP(H, Y ) asks: given an input digraph G and a homomorphism g : G → Y , does there exist a homomorphism f : G → H ? We prove that if H is hereditarily hard and Y H , then RHP(H, Y ) is NP-complete. Since non-bipartite graphs are hereditarily hard, this settles in the affirmative a conjecture of Hell and Nešetřil.
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