Coloring and Recognizing Mixed Interval Graphs

Authors Grzegorz Gutowski , Konstanty Junosza-Szaniawski , Felix Klesen , Paweł Rzążewski , Alexander Wolff , Johannes Zink

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Author Details

Grzegorz Gutowski
  • Theoretical Computer Science Department, Faculty of Mathematics and Computer Science, Jagiellonian University, Kraków, Poland
Konstanty Junosza-Szaniawski
  • Warsaw University of Technology, Poland
Felix Klesen
  • Universität Würzburg, Germany
Paweł Rzążewski
  • Warsaw University of Technology, Poland
  • Institute of Informatics, University of Warsaw, Poland
Alexander Wolff
  • Universität Würzburg, Germany
Johannes Zink
  • Universität Würzburg, Germany


We are indebted to Krzysztof Fleszar, Zbigniew Lonc, Karolina Okrasa, and Marta Piecyk for fruitful discussions. Additionally, we acknowledge the welcoming and productive atmosphere at the workshop Homonolo 2022, where some of the work was done.

Cite AsGet BibTex

Grzegorz Gutowski, Konstanty Junosza-Szaniawski, Felix Klesen, Paweł Rzążewski, Alexander Wolff, and Johannes Zink. Coloring and Recognizing Mixed Interval Graphs. In 34th International Symposium on Algorithms and Computation (ISAAC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 283, pp. 36:1-36:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


A mixed interval graph is an interval graph that has, for every pair of intersecting intervals, either an arc (directed arbitrarily) or an (undirected) edge. We are particularly interested in scenarios where edges and arcs are defined by the geometry of intervals. In a proper coloring of a mixed interval graph G, an interval u receives a lower (different) color than an interval v if G contains arc (u,v) (edge {u,v}). Coloring of mixed graphs has applications, for example, in scheduling with precedence constraints; see a survey by Sotskov [Mathematics, 2020]. For coloring general mixed interval graphs, we present a min {ω(G), λ(G)+1}-approximation algorithm, where ω(G) is the size of a largest clique and λ(G) is the length of a longest directed path in G. For the subclass of bidirectional interval graphs (introduced recently for an application in graph drawing), we show that optimal coloring is NP-hard. This was known for general mixed interval graphs. We introduce a new natural class of mixed interval graphs, which we call containment interval graphs. In such a graph, there is an arc (u,v) if interval u contains interval v, and there is an edge {u,v} if u and v overlap. We show that these graphs can be recognized in polynomial time, that coloring them with the minimum number of colors is NP-hard, and that there is a 2-approximation algorithm for coloring.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Graph theory
  • Interval Graphs
  • Mixed Graphs
  • Graph Coloring


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