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Towards the Recognition of Oriented Interval Graphs

Authors Lukas P. Bachmann , Jiří Fiala , Miriam Münch , Ignaz Rutter , Peter Stumpf , Alexander Wolff

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Subject Classification

ACM Subject Classification
  • Mathematics of computing → Graph theory
  • Theory of computation → Computational geometry
Keywords
  • Interval graphs
  • mixed graphs
  • oriented interval graphs
  • recognition

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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 φ : V(G) → {-1,1} of all intervals, (ii) a clique ordering σ, and (iii) a set E_cont ⊆ 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_cont or the orientation φ 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.

Cite As Get BibTex

Lukas P. Bachmann, Jiří Fiala, Miriam Münch, Ignaz Rutter, Peter Stumpf, and Alexander Wolff. Towards the Recognition of Oriented Interval Graphs. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 146:1-146:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.ESA.2026.146

Author Details

Lukas P. Bachmann
  • Universität Passau, Germany
Jiří Fiala
  • Charles University, Prague, Czech Republic
Miriam Münch
  • Universität Passau, Germany
Ignaz Rutter
  • Universität Passau, Germany
Peter Stumpf
  • Charles University, Prague, Czech Republic
Alexander Wolff
  • Universität Würzburg, Germany

Funding

  • Fiala, Jiří: Supported by grant no. 25-16847S of the Czech Science Foundation (GAČR).

References

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