Adjusted Interval Digraphs

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, we do not have a forbidden structure characterization nor a low-degree polynomial time 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. We relate adjusted interval digraphs to a list homomorphism problem. Each digraph H defines a corresponding list homomorphism problem L-HOM(H). We observe that if H is an adjusted interval digraph, then the problem L-HOM(H) is polynomial time solvable, and conjecture that for all other reflexive digraphs H the problem LHOM(H) is NP-complete. We present some preliminary evidence for the conjecture.