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# Vertex Deletion into Bipartite Permutation Graphs

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LIPIcs.IPEC.2020.5.pdf
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## 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.

## Cite As

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

## 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.

## Subject Classification

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

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