Adjusted Interval Digraphs and Complexity of List Homomorphisms

Interval digraphs were introduced by West et all. They can be recognized in polynomial time and admit a characterization in terms of incidence matrices. Nevertheless, they do not have a forbidden structure characterization nor a low-degree polynomial time recognition algorithm. We introduce a new class of ‘adjusted interval digraphs’, obtained by a slight change in the definition. By contrast, these digraphs have a natural forbidden structure characterization, parallel to a characterization for undirected graphs, and admit an easy recognition algorithm. Adjusted interval digraphs arise as natural analogues of interval graphs in the context of list homomorphism problems. Each digraph H defines a corresponding list homomorphism problem LHOM(H). For undirected graphs, it is known that reflexive interval graphs H yield polynomially solvable problems LHOM(H), while LHOM(H) is NP-complete for all other reflexive graphs H. We observe that if H is an adjusted interval digraph, then the problem LHOM(H) is also polynomial time solvable, and we conjecture that for all other reflexive digraphs H the problem LHOM(H) is NPcomplete. We prove the conjecture in two basic cases.