Interval graphs, adjusted interval digraphs, and reflexive list homomorphisms

Authors:
Tomás Feder;Pavol Hell;Jing Huang;Arash Rafiey
Affiliations:
68 Waverley Street, Palo Alto, CA 94301, United States;School of Computing Science, Simon Fraser University, Burnaby, BC, Canada V5A 1S6;Department of Mathematics and Statistics, University of Victoria, Victoria, B.C., Canada V8W 3R4;Istituto Dalle Molle di Studi sull'Intelligenza Artificiale, 6928 Manno-Lugano, Switzerland
Venue:
Discrete Applied Mathematics
Year:
2012

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Abstract

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 H is a reflexive undirected graph, the list homomorphism 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 that it is also NP-complete otherwise. We show that our results imply the conjecture in two important cases.