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Vertex Deletion into Bipartite Permutation Graphs

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

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

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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.

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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.

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