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Generalizing Roberts' Characterization of Unit Interval Graphs

Authors Virginia Ardévol Martínez , Romeo Rizzi, Abdallah Saffidine, Florian Sikora , Stéphane Vialette

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  • Mathematics of computing → Graph theory
Keywords
  • Interval graphs
  • Multiple Interval Graphs
  • Unit Interval Graphs
  • Characterization

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Abstract

For any natural number d, a graph G is a (disjoint) d-interval graph if it is the intersection graph of (disjoint) d-intervals, the union of d (disjoint) intervals on the real line. Two important subclasses of d-interval graphs are unit and balanced d-interval graphs (where every interval has unit length or all the intervals associated to a same vertex have the same length, respectively). A celebrated result by Roberts gives a simple characterization of unit interval graphs being exactly claw-free interval graphs. Here, we study the generalization of this characterization for d-interval graphs. In particular, we prove that for any d ⩾ 2, if G is a K_{1,2d+1}-free interval graph, then G is a unit d-interval graph. However, somehow surprisingly, under the same assumptions, G is not always a disjoint unit d-interval graph. This implies that the class of disjoint unit d-interval graphs is strictly included in the class of unit d-interval graphs. Finally, we study the relationships between the classes obtained under disjoint and non-disjoint d-intervals in the balanced case and show that the classes of disjoint balanced 2-intervals and balanced 2-intervals coincide, but this is no longer true for d > 2.

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Virginia Ardévol Martínez, Romeo Rizzi, Abdallah Saffidine, Florian Sikora, and Stéphane Vialette. Generalizing Roberts' Characterization of Unit Interval Graphs. In 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 306, pp. 12:1-12:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.MFCS.2024.12

Author Details

Virginia Ardévol Martínez
  • Université Paris-Dauphine, PSL University, CNRS, LAMSADE, 75016 Paris, France
Romeo Rizzi
  • Department of Computer Science, University of Verona, Italy
Abdallah Saffidine
  • University of New South Wales, Sydney, Australia
Florian Sikora
  • Université Paris-Dauphine, PSL University, CNRS, LAMSADE, 75016 Paris, France
Stéphane Vialette
  • LIGM, CNRS, Univ Gustave Eiffel, F77454 Marne-la-Vallée, France

Funding

This work was partially supported by the ANR project ANR-21-CE48-0022 ("S-EX-AP-PE-AL").

Acknowledgements

Part of this work was conducted when RR was an invited professor at Université Paris-Dauphine.

Supplementary Materials

References

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