, Volume 79, Issue 1, pp 66–95 | Cite as

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

  • Mamadou Moustapha Kanté
  • Eun Jung Kim
  • O-joung Kwon
  • Christophe Paul


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.


Linear rankwidth Rankwidth Cliquewidth Thread graph Necklace graph 



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


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Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  • Mamadou Moustapha Kanté
    • 1
  • Eun Jung Kim
    • 2
  • O-joung Kwon
    • 3
    • 4
  • Christophe Paul
    • 5
  1. 1.LIMOS, CNRS, Université Clermont AuvergneAubièreFrance
  2. 2.LAMSADECNRS - Université Paris DauphineParisFrance
  3. 3.Institute for Computer Science and ControlHungarian Academy of SciencesBudapestHungary
  4. 4.Institute of Software Technology and Theoretical Computer ScienceTechnische UniversitätBerlinGermany
  5. 5.LIRMMCNRS - Université MontpellierMontpellierFrance

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