Document Open Access Logo

Vertex Deletion into Bipartite Permutation Graphs

Authors Łukasz Bożyk , Jan Derbisz, Tomasz Krawczyk , Jana Novotná , Karolina Okrasa

PDF

File

Thumbnail PDF
PDF
LIPIcs.IPEC.2020.5.pdf
  • Filesize: 0.78 MB
  • 16 pages

Document Identifiers

Related Versions

Subject Classification

ACM Subject Classification
  • Theory of computation → Graph algorithms analysis
Keywords
  • permutation graphs
  • comparability graphs
  • partially ordered set
  • graph modification problems

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

Abstract

A permutation graph can be defined as an intersection graph of segments whose endpoints lie on two parallel lines 𝓁₁ and 𝓁₂, one on each. A bipartite permutation graph is a permutation graph which is bipartite. In this paper we study the parameterized complexity of the bipartite permutation vertex deletion problem, which asks, for a given n-vertex graph, whether we can remove at most k vertices to obtain a bipartite permutation graph. This problem is NP-complete by the classical result of Lewis and Yannakakis [John M. Lewis and Mihalis Yannakakis, 1980].
We analyze the structure of the so-called almost bipartite permutation graphs which may contain holes (large induced cycles) in contrast to bipartite permutation graphs. We exploit the structural properties of the shortest hole in a such graph. We use it to obtain an algorithm for the bipartite permutation vertex deletion problem with running time f(k)n^O(1), and also give a polynomial-time 9-approximation algorithm.

Cite As Get BibTex

Łukasz Bożyk, Jan Derbisz, Tomasz Krawczyk, Jana Novotná, and Karolina Okrasa. Vertex Deletion into Bipartite Permutation Graphs. In 15th International Symposium on Parameterized and Exact Computation (IPEC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 180, pp. 5:1-5:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020) https://doi.org/10.4230/LIPIcs.IPEC.2020.5

Author Details

Łukasz Bożyk
  • Faculty of Mathematics, Informatics and Mechanics, University of Warsaw, Poland
Jan Derbisz
  • Theoretical Computer Science Department, Faculty of Mathematics and Computer Science, Jagiellonian University, Kraków, Poland
Tomasz Krawczyk
  • Theoretical Computer Science Department, Faculty of Mathematics and Computer Science, Jagiellonian University, Kraków, Poland
Jana Novotná
  • Faculty of Mathematics, Informatics and Mechanics, University of Warsaw, Poland
  • Faculty of Mathematics and Physics, Charles University, Prague, Czech Republic
Karolina Okrasa
  • Faculty of Mathematics and Information Science, Warsaw University of Technology, Poland
  • Faculty of Mathematics, Informatics and Mechanics, University of Warsaw, Poland

Funding

Łukasz Bożyk, Jana Novotná, Karolina Okrasa: This research is supported by European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme, grant agreement 714704. It was partially carried out during the Parameterized Algorithms Retreat of the University of Warsaw, PARUW 2020, held in Krynica-Zdrój in February 2020.
  • Krawczyk, Tomasz: This research is supported by National Science Center of Poland, grant UMO-2015/17/B/ST6/01873.
  • Novotná, Jana: Supported by the student grant SVV–2020–260578.

Acknowledgements

The authors would like to thank Bartosz Walczak for valuable comments and help with merging two groups of researchers working on similar projects into one. They also thank to anonymous reviewers for helpful comments.

References

  1. Kirby A Baker, Peter C Fishburn, and Fred S Roberts. Partial orders of dimension 2. Networks, 2(1):11-28, 1972. Google Scholar
  2. Andreas Brandstädt and Dieter Kratsch. On the restriction of some NP-complete graph problems to permutation graphs. In Fundamentals of Computation Theory, FCT '85, Cottbus, GDR, September 9-13, 1985, volume 199 of Lecture Notes in Computer Science, pages 53-62. Springer, 1985. URL: https://doi.org/10.1007/BFb0028791.
  3. Leizhen Cai. Fixed-parameter tractability of graph modification problems for hereditary properties. Information Processing Letters, 58(4):171-176, 1996. Google Scholar
  4. Yixin Cao. Linear recognition of almost interval graphs. In Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2016, Arlington, VA, USA, January 10-12, 2016, pages 1096-1115. SIAM, 2016. URL: https://doi.org/10.1137/1.9781611974331.ch77.
  5. Yixin Cao. Unit interval editing is fixed-parameter tractable. Information and Computation, 253:109-126, 2017. Google Scholar
  6. Yixin Cao and Dániel Marx. Chordal editing is fixed-parameter tractable. Algorithmica, 75(1):118-137, 2016. Google Scholar
  7. Maria Chudnovsky, Neil Robertson, Paul Seymour, and Robin Thomas. The strong perfect graph theorem. Annals of Mathematics, pages 51-229, 2006. Google Scholar
  8. Jitender S. Deogun and George Steiner. Polynomial algorithms for hamiltonian cycle in cocomparability graphs. SIAM J. Comput., 23(3):520-552, 1994. URL: https://doi.org/10.1137/S0097539791200375.
  9. Jack Edmonds and Richard M Karp. Theoretical improvements in algorithmic efficiency for network flow problems. Journal of the ACM (JACM), 19(2):248-264, 1972. Google Scholar
  10. Tibor Gallai. Transitiv orientierbare graphen. Acta Mathematica Academiae Scientiarum Hungarica, 18(1-2):25-66, 1967. Google Scholar
  11. Martin Charles Golumbic, Doron Rotem, and Jorge Urrutia. Comparability graphs and intersection graphs. Discret. Math., 43(1):37-46, 1983. URL: https://doi.org/10.1016/0012-365X(83)90019-5.
  12. Martin Grötschel, László Lovász, and Alexander Schrijver. Geometric Algorithms and Combinatorial Optimization, volume 2 of Algorithms and Combinatorics. Springer, 1988. URL: https://doi.org/10.1007/978-3-642-97881-4.
  13. Pinar Heggernes, Pim van 't Hof, Bart M. P. Jansen, Stefan Kratsch, and Yngve Villanger. Parameterized complexity of vertex deletion into perfect graph classes. Theor. Comput. Sci., 511:172-180, 2013. URL: https://doi.org/10.1016/j.tcs.2012.03.013.
  14. Pinar Heggernes, Pim van 't Hof, Daniel Lokshtanov, and Jesper Nederlof. Computing the cutwidth of bipartite permutation graphs in linear time. SIAM J. Discret. Math., 26(3):1008-1021, 2012. URL: https://doi.org/10.1137/110830514.
  15. David Kelly. The 3-irreducible partially ordered sets. Can. J. of Math., 29:367-383, 1977. Google Scholar
  16. C Lekkeikerker and J Boland. Representation of a finite graph by a set of intervals on the real line. Fundamenta Mathematicae, 51(1):45-64, 1962. Google Scholar
  17. John M. Lewis and Mihalis Yannakakis. The node-deletion problem for hereditary properties is NP-complete. J. Comput. Syst. Sci., 20(2):219-230, 1980. URL: https://doi.org/10.1016/0022-0000(80)90060-4.
  18. Daniel Lokshtanov. Wheel-free deletion is W[2]-hard. In Martin Grohe and Rolf Niedermeier, editors, Parameterized and Exact Computation, Third International Workshop, IWPEC 2008, Victoria, Canada, May 14-16, 2008. Proceedings, volume 5018 of Lecture Notes in Computer Science, pages 141-147. Springer, 2008. URL: https://doi.org/10.1007/978-3-540-79723-4_14.
  19. Federico Mancini. Graph modification problems related to graph classes. PhD degree dissertation, University of Bergen Norway, 2, 2008. Google Scholar
  20. Dániel Marx. Chordal deletion is fixed-parameter tractable. Algorithmica, 57(4):747-768, 2010. URL: https://doi.org/10.1007/s00453-008-9233-8.
  21. Amir Pnueli, Abraham Lempel, and Shimon Even. Transitive orientation of graphs and identification of permutation graphs. Canadian Journal of Mathematics, 23(1):160-175, 1971. Google Scholar
  22. Jeremy P. Spinrad, Andreas Brandstädt, and Lorna Stewart. Bipartite permutation graphs. Discret. Appl. Math., 18(3):279-292, 1987. URL: https://doi.org/10.1016/S0166-218X(87)80003-3.
  23. William T. Trotter and John I. Moore Jr. Characterization problems for graphs, partially ordered sets, lattices, and families of sets. Discret. Math., 16(4):361-381, 1976. URL: https://doi.org/10.1016/S0012-365X(76)80011-8.
  24. Pim van 't Hof and Yngve Villanger. Proper interval vertex deletion. Algorithmica, 65(4):845-867, 2013. URL: https://doi.org/10.1007/s00453-012-9661-3.
  25. Yngve Villanger, Pinar Heggernes, Christophe Paul, and Jan Arne Telle. Interval completion is fixed parameter tractable. SIAM J. Comput., 38(5):2007-2020, 2009. URL: https://doi.org/10.1137/070710913.
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail
© 2023-2025 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact