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On Polynomial Kernelization of \(\mathcal {H}\)-free Edge Deletion

  • N. R. Aravind
  • R. B. SandeepEmail author
  • Naveen Sivadasan
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8894)

Abstract

For a set of graphs \(\mathcal {H}\), the \(\mathcal {H}\) -free Edge Deletion problem asks to find whether there exist at most \(k\) edges in the input graph whose deletion results in a graph without any induced copy of \(H\in \mathcal {H}\). In [3], it is shown that the problem is fixed-parameter tractable if \(\mathcal {H}\) is of finite cardinality. However, it is proved in [4] that if \(\mathcal {H}\) is a singleton set containing \(H\), for a large class of \(H\), there exists no polynomial kernel unless \(coNP\subseteq NP/poly\). In this paper, we present a polynomial kernel for this problem for any fixed finite set \(\mathcal {H}\) of connected graphs and when the input graphs are of bounded degree. We note that there are \(\mathcal {H}\) -free Edge Deletion problems which remain NP-complete even for the bounded degree input graphs, for example Triangle-free Edge Deletion [2] and Custer Edge Deletion( \(P_3\) -free Edge Deletion) [15]. When \(\mathcal {H}\) contains \(K_{1,s}\), we obtain a stronger result - a polynomial kernel for \(K_t\)-free input graphs (for any fixed \(t> 2\)). We note that for \(s>9\), there is an incompressibility result for \(K_{1,s}\) -free Edge Deletion for general graphs [5]. Our result provides first polynomial kernels for Claw-free Edge Deletion and Line Edge Deletion for \(K_t\)-free input graphs which are NP-complete even for \(K_4\)-free graphs [23] and were raised as open problems in [4, 19].

Keywords

Connected Graph Input Graph Polynomial Kernel Free Graph Edge Deletion 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • N. R. Aravind
    • 1
  • R. B. Sandeep
    • 1
    Email author
  • Naveen Sivadasan
    • 1
  1. 1.Department of Computer Science and EngineeringIndian Institute of Technology HyderabadHyderabadIndia

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