Complexity of list homomorphism problems for reflexive digraphs

For a fixed reflexive digraph H=(V(H), E(H)), the list homomorphim problem, LHOM(H), is the following: given an input graph G=(V(G), E(G)) and for each vertex v \in G a list L(v)\subset V(G), decide if there is a homomorphism f:G\longrightarrow H such that f(v)\in L(v) for each v\in V(G). This problem is equivalent to CSP(H_u), where H_u is the structure obtained by expanding H with all possible unary relations. Bulatov showed that a dichotomy holds for the LHOM problem of arbitrary structures, but his result does not tell us which classes of graphs are tractable and which ones are NP-complete. Feder, Hell, Huang and Rafiey showed that if H is an adjusted interval digraph then LHOM(H) is tractable and conjectured that otherwise it is NP-complete. We show that if H is not an adjusted interval digraph then it does not have weak-near-unanimity operations, and consequently it must be NP-complete. This is joint work with Arash Rafiey and Pavol Hell.