, Volume 79, Issue 1, pp 251–270 | Cite as

A Polynomial Kernel for Block Graph Deletion

  • Eun Jung Kim
  • O-Joung Kwon


In the Block Graph Deletion problem, we are given a graph G on n vertices and a positive integer k, and the objective is to check whether it is possible to delete at most k vertices from G to make it a block graph, i.e., a graph in which each block is a clique. In this paper, we obtain a kernel with \({\mathcal {O}}(k^{6})\) vertices for the Block Graph Deletion problem. This is a first step to investigate polynomial kernels for deletion problems into non-trivial classes of graphs of bounded rank-width, but unbounded tree-width. Our result also implies that Chordal Vertex Deletion admits a polynomial-size kernel on diamond-free graphs. For the kernelization and its analysis, we introduce the notion of ‘complete degree’ of a vertex. We believe that the underlying idea can be potentially applied to other problems. We also prove that the Block Graph Deletion problem can be solved in time \(10^{k}\cdot n^{{\mathcal {O}}(1)}\).


Block graph Block decomposition Kernelization algorithm Fixed parameter algorithm 


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© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.LAMSADECNRS - Université Paris DauphineParisFrance
  2. 2.Logic and SemanticsTechnische Universität BerlinBerlinGermany

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