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An FPT Algorithm and a Polynomial Kernel for Linear Rankwidth-1 Vertex Deletion

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Abstract

Linear rankwidth is a linearized variant of rankwidth, introduced by Oum and Seymour (J Comb Theory Ser B 96(4):514–528, 2006). Motivated from recent development on graph modification problems regarding classes of graphs of bounded treewidth or pathwidth, we study the Linear Rankwidth-1 Vertex Deletion problem (shortly, LRW1-Vertex Deletion). In the LRW1-Vertex Deletion problem, given an n-vertex graph G and a positive integer k, we want to decide whether there is a set of at most k vertices whose removal turns G into a graph of linear rankwidth at most 1 and find such a vertex set if one exists. While the meta-theorem of Courcelle, Makowsky, and Rotics implies that LRW1-Vertex Deletion can be solved in time \(f(k)\cdot n^3\) for some function f, it is not clear whether this problem allows a running time with a modest exponential function. We first establish that LRW1-Vertex Deletion can be solved in time \(8^k\cdot n^{{\mathcal {O}}(1)}\). The major obstacle to this end is how to handle a long induced cycle as an obstruction. To fix this issue, we define necklace graphs and investigate their structural properties. Later, we reduce the polynomial factor by refining the trivial branching step based on a cliquewidth expression of a graph, and obtain an algorithm that runs in time \(2^{{\mathcal {O}}(k)}\cdot n^4\). We also prove that the running time cannot be improved to \(2^{o(k)}\cdot n^{{\mathcal {O}}(1)}\) under the Exponential Time Hypothesis assumption. Lastly, we show that the LRW1-Vertex Deletion problem admits a polynomial kernel.

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Acknowledgments

O-joung Kwon would like to thank Sang-il Oum for suggesting the refined branching algorithm using cliquewidth.

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Author notes

  1. O-joung Kwon

    Present address: Institute of Software Technology and Theoretical Computer Science, Technische Universität, Berlin, Germany

Authors and Affiliations

  1. LIMOS, CNRS, Université Clermont Auvergne, Aubière, France

    Mamadou Moustapha Kanté

  2. LAMSADE, CNRS - Université Paris Dauphine, Paris, France

    Eun Jung Kim

  3. Institute for Computer Science and Control, Hungarian Academy of Sciences, Budapest, Hungary

    O-joung Kwon

  4. LIRMM, CNRS - Université Montpellier, Montpellier, France

    Christophe Paul

Authors
  1. Mamadou Moustapha Kanté
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  2. Eun Jung Kim
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  3. O-joung Kwon
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  4. Christophe Paul
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Corresponding author

Correspondence to O-joung Kwon.

Additional information

An extended abstract appeared in Proceedings of 10th International Symposium on Parameterized and Exact Computations, 2015 [30].

O-joung Kwon: supported by ERC Starting Grant PARAMTIGHT (No. 280152). The work was partially done while at Department of Mathematical Sciences, KAIST, and supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT and Future Planning (2011-0011653).

Christophe Paul: supported by the “Chercheur d’avenir – Languedoc-Roussillon” project KERNEL.

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Kanté, M.M., Kim, E.J., Kwon, Oj. et al. An FPT Algorithm and a Polynomial Kernel for Linear Rankwidth-1 Vertex Deletion. Algorithmica 79, 66–95 (2017). https://doi.org/10.1007/s00453-016-0230-z

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