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Simultaneous Representation of Proper and Unit Interval Graphs

Authors Ignaz Rutter , Darren Strash , Peter Stumpf , Michael Vollmer



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Author Details

Ignaz Rutter
  • Faculty of Computer Science and Mathematics, University of Passau, Germany
Darren Strash
  • Department of Computer Science, Hamilton College, Clinton, NY, USA
Peter Stumpf
  • Faculty of Computer Science and Mathematics, University of Passau, Germany
Michael Vollmer
  • Department of Informatics, Karlsruhe Institute of Technology (KIT), Germany

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Ignaz Rutter, Darren Strash, Peter Stumpf, and Michael Vollmer. Simultaneous Representation of Proper and Unit Interval Graphs. In 27th Annual European Symposium on Algorithms (ESA 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 144, pp. 80:1-80:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)
https://doi.org/10.4230/LIPIcs.ESA.2019.80

Abstract

In a confluence of combinatorics and geometry, simultaneous representations provide a way to realize combinatorial objects that share common structure. A standard case in the study of simultaneous representations is the sunflower case where all objects share the same common structure. While the recognition problem for general simultaneous interval graphs - the simultaneous version of arguably one of the most well-studied graph classes - is NP-complete, the complexity of the sunflower case for three or more simultaneous interval graphs is currently open. In this work we settle this question for proper interval graphs. We give an algorithm to recognize simultaneous proper interval graphs in linear time in the sunflower case where we allow any number of simultaneous graphs. Simultaneous unit interval graphs are much more "rigid" and therefore have less freedom in their representation. We show they can be recognized in time O(|V|*|E|) for any number of simultaneous graphs in the sunflower case where G=(V,E) is the union of the simultaneous graphs. We further show that both recognition problems are in general NP-complete if the number of simultaneous graphs is not fixed. The restriction to the sunflower case is in this sense necessary.

Subject Classification

ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
Keywords
  • Intersection Graphs
  • Recognition Algorithm
  • Proper/Unit Interval Graphs
  • Simultaneous Representations

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References

  1. Patrizio Angelini, Giordano Da Lozzo, and Daniel Neuwirth. On Some NP-complete SEFE Problems. In Sudebkumar Prasant Pal and Kunihiko Sadakane, editors, Algorithms and Computation: 8th International Workshop, WALCOM 2014, Chennai, Proceedings, pages 200-212. Springer, 2014. URL: https://doi.org/10.1007/978-3-319-04657-0_20.
  2. Thomas Bläsius, Stephen G. Kobourov, and Ignaz Rutter. Simultaneous Embedding of Planar Graphs. CoRR, abs/1204.5853, 2012. URL: http://arxiv.org/abs/1204.5853.
  3. Thomas Bläsius and Ignaz Rutter. Simultaneous PQ-Ordering with Applications to Constrained Embedding Problems. ACM Trans. Algorithms, 12(2):16:1-16:46, 2015. URL: https://doi.org/10.1145/2738054.
  4. Jan Bok and Nikola Jedličková. A note on simultaneous representation problem for interval and circular-arc graphs. arXiv preprint, 2018. URL: http://arxiv.org/abs/1811.04062.
  5. Kellogg S. Booth. PQ Tree Algorithms. PhD thesis, University of California, Berkeley, 1975. Google Scholar
  6. Kellogg S. Booth and George S. Lueker. Testing for the consecutive ones property, interval graphs, and graph planarity using PQ-tree algorithms. Journal of Computer and System Sciences, 13(3):335-379, 1976. URL: https://doi.org/10.1016/S0022-0000(76)80045-1.
  7. Peter Brass, Eowyn Cenek, Cristian A Duncan, Alon Efrat, Cesim Erten, Dan P Ismailescu, Stephen G Kobourov, Anna Lubiw, and Joseph SB Mitchell. On simultaneous planar graph embeddings. Computational Geometry, 36(2):117-130, 2007. URL: https://doi.org/10.1016/j.comgeo.2006.05.006.
  8. Steven Chaplick, Radoslav Fulek, and Pavel Klavík. Extending Partial Representations of Circle Graphs. In Stephen Wismath and Alexander Wolff, editors, Graph Drawing: 21st International Symposium, GD 2013, Bordeaux, Revised Selected Papers, pages 131-142. Springer, 2013. URL: https://doi.org/10.1007/978-3-319-03841-4_12.
  9. Celina M. Herrera de Figueiredo, João Meidanis, and Célia Picinin de Mello. A linear-time algorithm for proper interval graph recognition. Information Processing Letters, 56(3):179-184, 1995. URL: https://doi.org/10.1016/0020-0190(95)00133-W.
  10. Xiaotie Deng, Pavol Hell, and Jing Huang. Linear-Time Representation Algorithms for Proper Circular-Arc Graphs and Proper Interval Graphs. SIAM J. Comput., 25(2):390-403, 1996. URL: https://doi.org/10.1137/S0097539792269095.
  11. Alejandro Estrella-Balderrama, Elisabeth Gassner, Michael Jünger, Merijam Percan, Marcus Schaefer, and Michael Schulz. Simultaneous Geometric Graph Embeddings. In Seok-Hee Hong, Takao Nishizeki, and Wu Quan, editors, Graph Drawing: 15th International Symposium, GD 2007, Sydney. Revised Papers, pages 280-290. Springer, 2008. URL: https://doi.org/10.1007/978-3-540-77537-9_28.
  12. Elisabeth Gassner, Michael Jünger, Merijam Percan, Marcus Schaefer, and Michael Schulz. Simultaneous Graph Embeddings with Fixed Edges. In Fedor V. Fomin, editor, Graph-Theoretic Concepts in Computer Science: 32nd International Workshop, WG 2006, Bergen, Revised Papers, pages 325-335. Springer, 2006. URL: https://doi.org/10.1007/11917496_29.
  13. Paul W. Goldberg, Martin C. Golumbic, Haim Kaplan, and Ron Shamir. Four strikes against physical mapping of DNA. Journal of Computational Biology, 2(1):139-152, 1995. URL: https://doi.org/10.1089/cmb.1995.2.139.
  14. Martin Charles Golumbic. Algorithmic Graph Theory and Perfect Graphs (Annals of Discrete Mathematics, Vol 57). North-Holland Publishing Co., 2004. Google Scholar
  15. Martin Charles Golumbic, Haim Kaplan, and Ron Shamir. Graph sandwich problems. Journal of Algorithms, 19(3):449-473, 1995. URL: https://doi.org/10.1006/jagm.1995.1047.
  16. Pinar Heggernes, Pim van 't Hof, Daniel Meister, and Yngve Villanger. Induced Subgraph Isomorphism on proper interval and bipartite permutation graphs. Theoretical Computer Science, 562:252-269, 2015. URL: https://doi.org/10.1016/j.tcs.2014.10.002.
  17. Pavol Hell, Ron Shamir, and Roded Sharan. A Fully Dynamic Algorithm for Recognizing and Representing Proper Interval Graphs. SIAM J. Comput., 31(1):289-305, 2002. URL: https://doi.org/10.1137/S0097539700372216.
  18. Krishnam Raju Jampani and Anna Lubiw. Simultaneous Interval Graphs. In Otfried Cheong, Kyung-Yong Chwa, and Kunsoo Park, editors, Algorithms and Computation: 21st International Symposium, ISAAC 2010, Jeju Island, Proceedings, Part I, pages 206-217. Springer, 2010. URL: https://doi.org/10.1007/978-3-642-17517-6_20.
  19. Krishnam Raju Jampani and Anna Lubiw. The Simultaneous Representation Problem for Chordal, Comparability and Permutation Graphs. Journal of Graph Algorithms and Applications, 16(2):283-315, 2012. URL: https://doi.org/10.7155/jgaa.00259.
  20. Pavel Klavík, Jan Kratochvíl, Yota Otachi, Ignaz Rutter, Toshiki Saitoh, Maria Saumell, and Tomáš Vyskočil. Extending Partial Representations of Proper and Unit Interval Graphs. Algorithmica, 77(4):1071-1104, April 2017. URL: https://doi.org/10.1007/s00453-016-0133-z.
  21. Peter J Looges and Stephan Olariu. Optimal greedy algorithms for indifference graphs. Computers & Mathematics with Applications, 25(7):15-25, 1993. URL: https://doi.org/10.1016/0898-1221(93)90308-I.
  22. Ross M McConnell and Yahav Nussbaum. Linear-time recognition of probe interval graphs. In Amos Fiat and Peter Sanders, editors, Proceedings of the 17th Annual European Symposium on Algorithms (ESA'09), volume 5757 of Lecture Notes in Computer Science, pages 349-360. Springer, 2009. URL: https://doi.org/10.1007/978-3-642-04128-0_32.
  23. Yahav Nussbaum. Recognition of probe proper interval graphs. Discrete Applied Mathematics, 167:228-238, 2014. URL: https://doi.org/10.1016/j.dam.2013.11.013.
  24. Fred S Roberts. Representations of indifference relations. PhD thesis, Department of Mathematics, Stanford University, 1968. Google Scholar
  25. Fred S. Roberts. Indifference Graphs. In F. Harary, editor, Proof Techniques in Graph Theory, pages 139-146. Academic Press, New York, 1969. Google Scholar
  26. Marcus Schaefer. Toward a Theory of Planarity: Hanani-Tutte and Planarity Variants. In Walter Didimo and Maurizio Patrignani, editors, Graph Drawing: 20th International Symposium, GD 2012, Redmond, Revised Selected Papers, pages 162-173. Springer, 2013. URL: https://doi.org/10.1007/978-3-642-36763-2_15.
  27. Dale J. Skrien. A relationship between triangulated graphs, comparability graphs, proper interval graphs, proper circular-arc graphs, and nested interval graphs. Journal of Graph Theory, 6(3):309-316, 1982. URL: https://doi.org/10.1002/jgt.3190060307.
  28. Jeremy P. Spinrad. Efficient Graph Representations. Fields Institute Monographs. AMS, 2003. Google Scholar
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