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 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 we call an ‘invertible pair’. Our characterization yields an easy recognition algorithm of adjusted interval digraphs. It turns out 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. As a consequence, we derive both of these theorems from our results. In addition, adjusted interval digraphs naturally arise in the context of list homomorphism problems LHOM(H). If H is a reflexive undirected graph, the problem LHOM(H) is polynomial if H is an interval graph, and NP-complete otherwise. If H is a reflexive digraph, LHOM(H) is polynomial if H is an adjusted interval graph, and we conjecture it is also NP-complete otherwise. We show that our results imply the conjecture in two important cases.