Interval graphs, adjusted interval digraphs, and reflexive list homomorphisms

Interval graphs admit linear-time recognition algorithms and have several elegant forbidden structure characterizations. Interval digraphs can also be recognized in polynomial time, and they admit a characterization in terms of incidence matrices. Nevertheless, they do not have a known forbidden structure characterization or low-degree polynomial-time recognition algorithm.We introduce a new class of ‘adjusted interval digraphs’. By contrast, for these digraphs we exhibit a natural forbidden structure characterization, in terms of a novel structure which we call an ‘invertible pair’. Our characterization also yields a low-degree polynomial-time recognition algorithm of adjusted interval digraphs.It turns out that invertible pairs are also useful for undirected interval graphs, and our result yields a new forbidden structure characterization of interval graphs. In fact, it can be shown to be a natural link proving the equivalence of some known characterizations of interval graphs—the theorems of Lekkerkerker and Boland, and of Fulkerson and Gross.In addition, adjusted interval digraphs naturally arise in the context of list homomorphism problems. If HH is a reflexive undirected graph, the list homomorphism problem LHOM(HH) is polynomial if HH is an interval graph, and NP-complete otherwise. If HH is a reflexive digraph, LHOM(HH) is polynomial if HH is an adjusted interval graph, and we conjecture that it is also NP-complete otherwise. We show that our results imply the conjecture in two important cases.