eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2024-08-23
12:1
12:15
10.4230/LIPIcs.MFCS.2024.12
article
Generalizing Roberts' Characterization of Unit Interval Graphs
Ardévol Martínez, Virginia
1
https://orcid.org/0000-0002-3703-2335
Rizzi, Romeo
2
Saffidine, Abdallah
3
Sikora, Florian
1
https://orcid.org/0000-0003-2670-6258
Vialette, Stéphane
4
https://orcid.org/0000-0003-2308-6970
Université Paris-Dauphine, PSL University, CNRS, LAMSADE, 75016 Paris, France
Department of Computer Science, University of Verona, Italy
University of New South Wales, Sydney, Australia
LIGM, CNRS, Univ Gustave Eiffel, F77454 Marne-la-Vallée, France
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.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol306-mfcs2024/LIPIcs.MFCS.2024.12/LIPIcs.MFCS.2024.12.pdf
Interval graphs
Multiple Interval Graphs
Unit Interval Graphs
Characterization