Recognition and Characterization of Chronological Interval Digraphs

Interval graphs admit elegant structural characterizations and linear time recognition algorithms; on the other hand, the usual interval digraphs lack a forbidden structure characterization as well as a low-degree polynomial time recognition algorithm. In this paper we identify another natural digraph analogue of interval graphs that we call ”chronological interval digraphs”. By contrast, the new class admits both a forbidden structure characterization and 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. Specifically, $G$ is a chronological interval digraph if there exists a family of closed intervals $I_ v$ , $v \in V(G)$, such that $uv$ is an arc of $G$ if and only if $I_ u$  intersects $I_ v$  and the left endpoint of $I_ u$  is not greater than the left endpoint of $I_ v$ . (Equivalently, if and only if $I_ u$  contains the left endpoint of $I_ v$ .) We characterize chronological interval digraphs in terms of vertex orderings, in terms of forbidden substructures, and in terms of a novel structure of so-called $Q$-paths. The first two characterizations exhibit strong similarity with the corresponding characterizations of interval graphs. The last characterization leads to a linear time recognition algorithm.