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# Generalizing Roberts' Characterization of Unit Interval Graphs

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LIPIcs.MFCS.2024.12.pdf
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## Acknowledgements

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

## Cite As

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

## 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.

## Subject Classification

##### ACM Subject Classification
• Mathematics of computing → Graph theory
##### Keywords
• Interval graphs
• Multiple Interval Graphs
• Unit Interval Graphs
• Characterization

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## References

1. Virginia Ardévol Martínez, Romeo Rizzi, Florian Sikora, and Stéphane Vialette. Recognizing unit multiple intervals is hard. In Satoru Iwata and Naonori Kakimura, editors, 34th International Symposium on Algorithms and Computation, ISAAC 2023, volume 283 of LIPIcs, pages 8:1-8:18. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2023. URL: https://doi.org/10.4230/LIPICS.ISAAC.2023.8.
2. Reuven Bar-Yehuda, Magnús M Halldórsson, Joseph Naor, Hadas Shachnai, and Irina Shapira. Scheduling split intervals. SIAM J. Comput., 36(1):1-15, 2006.
3. Kenneth P. Bogart and Douglas B. West. A short proof that 'proper = unit'. Discret. Math., 201(1-3):21-23, 1999. URL: https://doi.org/10.1016/S0012-365X(98)00310-0.
4. Kellogg S. Booth and George S. Lueker. Testing for the consecutive ones property, interval graphs, and graph planarity using PQ-Tree algorithms. J. Comput. Syst. Sci., 13(3):335-379, 1976. URL: https://doi.org/10.1016/S0022-0000(76)80045-1.
5. Ayelet Butman, Danny Hermelin, Moshe Lewenstein, and Dror Rawitz. Optimization problems in multiple-interval graphs. ACM Trans. Algorithms, 6(2):1-18, 2010.
6. Derek G. Corneil, Stephan Olariu, and Lorna Stewart. The LBFS structure and recognition of interval graphs. SIAM J. Discret. Math., 23(4):1905-1953, 2009. URL: https://doi.org/10.1137/S0895480100373455.
7. Maxime Crochemore, Danny Hermelin, Gad M. Landau, Dror Rawitz, and Stéphane Vialette. Approximating the 2-interval pattern problem. Theor. Comput. Sci., 395(2-3):283-297, 2008. URL: https://doi.org/10.1016/j.tcs.2008.01.007.
8. Guillermo Durán, Florencia Fernández Slezak, Luciano N. Grippo, Fabiano de S. Oliveira, and Jayme Luiz Szwarcfiter. On unit d–interval graphs. VII Latin American Workshop on Cliques in Graphs, 2016. URL: https://www.mate.unlp.edu.ar/~liliana/lawclique_2016/ffslezak.pdf, https://www.mate.unlp.edu.ar/~liliana/lawclique_2016/prolist.pdf.
9. Paul Erdös and Douglas B West. A note on the interval number of a graph. Discret. Math., 55(2):129-133, 1985.
10. Michael R. Fellows, Danny Hermelin, Frances A. Rosamond, and Stéphane Vialette. On the parameterized complexity of multiple-interval graph problems. Theor. Comput. Sci., 410(1):53-61, 2009. URL: https://doi.org/10.1016/j.tcs.2008.09.065.
11. Peter C. Fishburn. Interval Orders and Interval Graphs: A Study of Partially Ordered Sets. Wiley, 1985.
12. Mathew C. Francis, Daniel Gonçalves, and Pascal Ochem. The maximum clique problem in multiple interval graphs. Algorithmica, 71(4):812-836, 2015. URL: https://doi.org/10.1007/s00453-013-9828-6.
13. Philippe Gambette and Stéphane Vialette. On restrictions of balanced 2-interval graphs. In Andreas Brandstädt, Dieter Kratsch, and Haiko Müller, editors, Graph-Theoretic Concepts in Computer Science, 33rd International Workshop, WG 2007, Dornburg, Germany, June 21-23, 2007. Revised Papers, volume 4769 of LNCS, pages 55-65. Springer, 2007. URL: https://doi.org/10.1007/978-3-540-74839-7_6.
14. Frédéric Gardi. The Roberts characterization of proper and unit interval graphs. Discret. Math., 307(22):2906-2908, 2007. URL: https://doi.org/10.1016/j.disc.2006.04.043.
15. Jerrold R. Griggs and Douglas B. West. Extremal values of the interval number of a graph. SIAM J. Algebraic Discret. Methods, 1(1):1-7, 1980. URL: https://doi.org/10.1137/0601001.
16. Michel Habib, Ross M. McConnell, Christophe Paul, and Laurent Viennot. Lex-BFS and partition refinement, with applications to transitive orientation, interval graph recognition and consecutive ones testing. Theor. Comput. Sci., 234(1-2):59-84, 2000. URL: https://doi.org/10.1016/S0304-3975(97)00241-7.
17. Minghui Jiang. On the parameterized complexity of some optimization problems related to multiple-interval graphs. Theor. Comput. Sci., 411(49):4253-4262, 2010. URL: https://doi.org/10.1016/j.tcs.2010.09.001.
18. Minghui Jiang. Recognizing d-interval graphs and d-track interval graphs. Algorithmica, 66(3):541-563, 2013. URL: https://doi.org/10.1007/s00453-012-9651-5.
19. Minghui Jiang and Yong Zhang. Parameterized complexity in multiple-interval graphs: Domination, partition, separation, irredundancy. Theor. Comput. Sci., 461:27-44, 2012. URL: https://doi.org/10.1016/j.tcs.2012.01.025.
20. Deborah Joseph, Joao Meidanis, and Prasoon Tiwari. Determining DNA sequence similarity using maximum independent set algorithms for interval graphs. In Scandinavian Workshop on Algorithm Theory, pages 326-337. Springer, 1992.
21. C Lekkerkerker and Johan Boland. Representation of a finite graph by a set of intervals on the real line. Fundamenta Mathematicae, 51(1):45-64, 1962.
22. Robert McGuigan. Presentation at NSF-CBMS Conference at Colby College, 1977.
23. Terry A McKee and Fred R McMorris. Topics in intersection graph theory. SIAM, 1999.
24. Fred S. Roberts. Indifference graphs. In F. Harary, editor, Proof Techniques in Graph Theory, pages 139-146. Academic Press, NY, 1969.
25. Fred S. Roberts. Graph theory and its applications to problems of society. SIAM, 1978.
26. Abdallah Saffidine. Unit 2-interval graph checker. Software, swhId: https://archive.softwareheritage.org/swh:1:dir:3d7b6495d1f70618a537cd23c94530c23c030215;origin=https://github.com/AbdallahS/unit-graphs;visit=swh:1:snp:84b7b457760a919cc007e2290179e1fc6fe861e3;anchor=swh:1:rev:516ab210d2ff334ac34348619bc42d252824cac4 (visited on 2024-08-06). URL: https://github.com/AbdallahS/unit-graphs.
27. Edward R Scheinerman and Douglas B West. The interval number of a planar graph: Three intervals suffice. J. Comb. Theory, Ser. B, 35(3):224-239, 1983.
28. Alexandre Simon. Algorithmic study of 2-interval graphs. Master’s thesis, Delft University of Technology, 2021.
29. William T. Trotter and Frank Harary. On double and multiple interval graphs. J. Graph Theory, 3(3):205-211, 1979. URL: https://doi.org/10.1002/jgt.3190030302.
30. Stéphane Vialette. On the computational complexity of 2-interval pattern matching problems. Theor. Comput. Sci., 312(2-3):223-249, 2004. URL: https://doi.org/10.1016/j.tcs.2003.08.010.
31. Douglas B. West and David B. Shmoys. Recognizing graphs with fixed interval number is NP-complete. Discret. Appl. Math., 8(3):295-305, 1984. URL: https://doi.org/10.1016/0166-218X(84)90127-6.
32. Kazuaki Yamazaki, Toshiki Saitoh, Masashi Kiyomi, and Ryuhei Uehara. Enumeration of nonisomorphic interval graphs and nonisomorphic permutation graphs. Theor. Comput. Sci., 806:310-322, 2020. URL: https://doi.org/10.1016/J.TCS.2019.04.017.