On the restricted homomorphism problem

Abstract

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.