Towards the Recognition of Oriented Interval Graphs

2026-07-06

AI Summary

This study addresses the recognition problem for oriented interval graphs: given a mixed graph, determine whether it can be represented as an intersection graph of intervals with left-to-right orientations, where overlapping intervals in the same direction correspond to directed arcs, and overlapping or nested intervals in opposite directions correspond to undirected edges. By uncovering the mutual constraints among interval orientations, clique orders, and containment edges, the work provides the first combinatorial characterizations—specifying the set of feasible orientations under fixed containment edges and the admissible clique orders under fixed orientations. Leveraging these characterizations, the authors devise linear-time recognition algorithms for two restricted cases, including proper and unit oriented interval graphs, significantly improving upon existing quadratic-time approaches.

Abstract

Oriented interval graphs, a recent generalization of interval graphs introduced by Gutowski et al. [GD 2022], are intersection graphs of intervals, each of which is oriented either left or right. Such a representation defines a mixed intersection graph: overlapping intervals with the same orientation define a (directed) arc; nested intervals (irrespective of the orientations of the intervals) and overlapping intervals of opposite orientations define an (undirected) edge. An oriented interval representation of a mixed graph $G$ can be described combinatorially by the combination of (i) an orientation $\varphi \colon V(G) \to \{-1,1\}$ of all intervals, (ii) a clique ordering $σ$, and (iii) a set $E_\mathrm{cont} \subseteq E(G)$ of containment edges, which are represented by nested intervals. The non-trivial dependencies between these three ingredients make the recognition of oriented interval graphs a challenging problem. In this paper, we take steps towards a general recognition algorithm by studying how orientation, clique ordering, and containment edges influence and restrict each other. We characterize the orientations that are consistent with a given set of containment edges as well as the clique orderings that are consistent with a given orientation. Based on these characterizations, we give linear-time algorithms for two constrained versions of the recognition problem where, in addition to the mixed input graph $G$, either the set of containment edges $E_\mathrm{cont}$ or the orientation $\varphi$ is prescribed. This improves a quadratic-time algorithm of Gutowski et al. for the case that all vertices have the same orientation; an assumption that determines both the orientation and the containment edges. In particular, this also solves the recognition problem for oriented proper (or unit) interval graphs.

Problem

• oriented interval graphs
• recognition problem
• mixed graphs
• interval representation
• containment edges

Innovation

• oriented interval graphs
• recognition algorithm
• containment edges
• clique ordering
• linear-time algorithm