On homomorphisms to acyclic local tournaments

A homomorphism of a digraph to another digraph is an edgepreserving vertex mapping. A local tournament is a digraph in which the inset as well as the outset of each vertex induces a tournament. Thus acyclic local tournaments generalize both directed paths and transitive tournaments. In both these cases there is a simple characterization of homomorphic preimages. Namely, if H is a directed path, or a transitive tournament, then G admits a homomorphism to H if and only if each oriented path which admits a homomorphism to G also admits a homomorphism to H . We prove that this result holds for all acyclic local tournaments. © 1995 John Wiley & Sons, Inc.