Recognition and characterization of chronological interval digraphs

Abstract

Interval graphs admit elegant ordering and structural characterizations as well as linear time recognition algorithms. On the other hand, the usual interval digraphs lack all three of these characteristics. In this paper we identify another natural digraph analogue of interval graphs that we call "chronological interval digraphs". By contrast, the new class admits an ordering characterization, several forbidden substructure characterizations, as well as a linear time recognition algorithm. Chronological interval digraphs arise by interpreting the standard definition of an interval graph with a natural orientation of its edges: G is a chronological interval digraph if there exists a family of closed intervals Iv, v ∈ V (G), such that uv is an arc of G if and only if Iu contains the left endpoint of Iv. We characterize chronological interval digraphs in terms of vertex orderings, and in terms of two kinds of forbidden substructures. These characterizations exhibit strong similarity with the corresponding characterizations of interval graphs, and lead to a linear time recognition algorithm.