For a fixed oriented tree T, we consider the complexity of deciding whether or not a given digraph G is homomorphic to T. It was shown by Gutjahr, Woeginger and Welzl that there exist trees T for which this homomorphism problem is NP-complete. However, it seems difficult to decide just which trees T yield NP-complete homomorphism problems. In this paper, we first identify a class of simple trees with NP-complete homomorphism problems; these trees have exactly one vertex of degree 3 and all other vertices of degree 1 or 2. Our smallest tree has only 45 vertices. (The previous known smallest NP-complete tree has 81 vertices.) In order to gain insight into the structure of oriented trees T which have NP-complete homomorphism problems, we list all subtrees that are necessary in such oriented trees.
Hell, P., Nešetřil, J., & Zhu, X. (1996). Complexity of tree homomorphisms. Discrete Applied Mathematics, 70(1), 23–36. https://doi.org/10.1016/0166-218X(96)00099-6