Parameterized Vertex Deletion Problems for Hereditary Graph Classes with a Block Property

  • Édouard Bonnet
  • Nick Brettell
  • O-joung Kwon
  • Dániel Marx
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9941)


For a class of graphs \(\mathcal {P}\), the Bounded \(\mathcal {P}\)-Block Vertex Deletion problem asks, given a graph G on n vertices and positive integers k and d, whether there is a set S of at most k vertices such that each block of \(G-S\) has at most d vertices and is in \(\mathcal {P}\). We show that when \(\mathcal {P}\) satisfies a natural hereditary property and is recognizable in polynomial time, Bounded \(\mathcal {P}\)-Block Vertex Deletion can be solved in time \(2^{\mathcal {O}(k \log d)}n^{\mathcal {O}(1)}\), and this running time cannot be improved to \(2^{o(k \log d)}n^{\mathcal {O}(1)}\), in general, unless the Exponential Time Hypothesis fails. On the other hand, if \(\mathcal {P}\) consists of only complete graphs, or only \(K_1, K_2\), and cycle graphs, then Bounded \(\mathcal {P}\)-Block Vertex Deletion admits a \(c^{k}n^{\mathcal {O}(1)}\)-time algorithm for some constant c independent of d. We also show that Bounded \(\mathcal {P}\)-Block Vertex Deletion admits a kernel with \(\mathcal {O}(k^2 d^7)\) vertices.


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Copyright information

© Springer-Verlag GmbH Germany 2016

Authors and Affiliations

  • Édouard Bonnet
    • 1
  • Nick Brettell
    • 1
  • O-joung Kwon
    • 1
  • Dániel Marx
    • 1
  1. 1.Institute for Computer Science and ControlHungarian Academy of Sciences (MTA SZTAKI)BudapestHungary

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