List Homomorphisms to Reflexive Digraphs

We study list homomorphism problems L-HOM(H) for the class of reflexive digraphs H (digraphs in which each vertex has a loop). These problems have been intensively studied in the case of undirected graphs H, and appear to be more difficult for digraphs. However, it is known that each problem L-HOM(H) is NP-complete or polynomial time solvable. In this paper we focus on reflexive digraphs. We introduce a new class of ‘adjusted interval digraphs’, point out that the list homomorphism problem L-HOM(H) is polynomial time solvable when H is an adjusted interval digraph, and conjecture it is NP-complete otherwise. It suffices to prove the conjecture for digraphs H for which the underlying graph is an interval graph, and we prove the conjecture when the underlying graph of H is a clique or a tree, thus establishing a possible basis for the general result. The class of adjusted interval digraphs appears interesting in its own right, and we provide a forbidden substructure characterization which implies a polynomial time recognition algorithm.