List homomorphisms and retractions to reflexive digraphs

We study the list homomorphism and retraction problems for the class of reflexive digraphs (digraphs in which each vertex has a loop). These problems have been intensively studied in the case of undirected graphs, but the situation seems more complex for digraphs. We also focus on an intermediate ‘subretraction’ problem. It turns out that the complexity of the subretraction problem can be classified at least for large classes of reflexive digraphs; by contrast, the complexity of the retraction problem for reflexive digraphs seems difficult to classify. For general list homomorphism problems, we conjecture that the problem is NP-complete unless H is an ‘adjusted’ interval digraph, in which case it is polynomial time solvable. We prove several cases of this conjecture. The class of adjusted interval digraphs appears interesting in its own right.