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Mim-Width II. The Feedback Vertex Set Problem

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Abstract

We give a first polynomial-time algorithm for (Weighted) Feedback Vertex Set on graphs of bounded maximum induced matching width (mim-width). Explicitly, given a branch decomposition of mim-width w, we give an \(n^{\mathcal {O}(w)}\)-time algorithm that solves Feedback Vertex Set. This provides a unified polynomial-time algorithm for many well-known classes, such as Interval graphs, Permutation graphs, and Leaf power graphs (given a leaf root), and furthermore, it gives the first polynomial-time algorithms for other classes of bounded mim-width, such as Circular Permutation and Circular k-Trapezoid graphs (given a circular k-trapezoid model) for fixed k. We complement our result by showing that Feedback Vertex Set is \(\textsf {W}[1]\)-hard when parameterized by w and the hardness holds even when a linear branch decomposition of mim-width w is given.

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Notes

  1. A cut of a graph is a bipartition of its vertex set.

  2. Given a (circular) k-trapezoid model.

  3. It is known that powers of permutation graphs are not necessarily permutation graphs [4, 15].

  4. Note however that in contrast to the previously mentioned classes, for Leaf Power graphs it is currently not known whether the corresponding decomposition can be computed in polynomial time. The construction in the proof presented in [23] uses a given leaf root of the input graph and it is still not known whether a leaf root of a leaf power graph can be computed in polynomial time.

  5. i.e. The vertices in B that have neighbors in A.

  6. We would like to stress that the reduction given here is closely inspired by the one due to Fomin, Golovach and Raymond [16]. The main difference in the construction of H and the resulting H-graph \(G'\) revolves around introducing the new vertices to H in Steps 3 and 4 below which are key to fit the reduction for Maximum Induced Forest. Note also that the subdivisions described below are the same as in [16].

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Authors and Affiliations

  1. Department of Informatics, University of Bergen, Bergen, Norway

    Lars Jaffke & Jan Arne Telle

  2. Department of Mathematics, Incheon National University, Incheon, South Korea

    O-joung Kwon

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  1. Lars Jaffke
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  2. O-joung Kwon
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  3. Jan Arne Telle
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Correspondence to Lars Jaffke.

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The work was partially done while the authors were at Polytechnic University of Valencia, Spain. Based on an extended abstract that appeared at STACS 2018 [23] and the note [20]. The first part of this series, titled “Mim-Width I. Induced Path Problems” [22], is based on an extended abstract that appeared at IPEC 2017 [21]. L. J. is supported by the Bergen Research Foundation (BFS). O. K. is supported by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (ERC consolidator Grant DISTRUCT, Agreement No. 648527), and also supported by the National Research Foundation of Korea (NRF) Grant funded by the Ministry of Education (No. NRF-2018R1D1A1B07050294).

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Jaffke, L., Kwon, Oj. & Telle, J.A. Mim-Width II. The Feedback Vertex Set Problem. Algorithmica 82, 118–145 (2020). https://doi.org/10.1007/s00453-019-00607-3

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