Algorithmica

, Volume 71, Issue 4, pp 989–1006

Faster Parameterized Algorithms for Deletion to Split Graphs

  • Esha Ghosh
  • Sudeshna Kolay
  • Mrinal Kumar
  • Pranabendu Misra
  • Fahad Panolan
  • Ashutosh Rai
  • M. S. Ramanujan
Article

DOI: 10.1007/s00453-013-9837-5

Cite this article as:
Ghosh, E., Kolay, S., Kumar, M. et al. Algorithmica (2015) 71: 989. doi:10.1007/s00453-013-9837-5

Abstract

An undirected graph is said to be split if its vertex set can be partitioned into two sets such that the subgraph induced on one of them is a complete graph and the subgraph induced on the other is an independent set. We initiate a systematic study of parameterized complexity of the problem of deleting the minimum number of vertices or edges from a given input graph so that the resulting graph is split. We give efficient fixed-parameter algorithms and polynomial sized kernels for the problem. More precisely,
  1. 1.

    for Split Vertex Deletion, the problem of determining whether there are k vertices whose deletion results in a split graph, we give an \({\mathcal{O}}^{*}(2^{k})\) algorithm (\({\mathcal{O}}^{*}()\) notation hides factors that are polynomial in the input size) improving on the previous best bound of \({\mathcal{O}}^{*} (2.32^{k})\). We also give an \({\mathcal{O}}(k^{3})\)-sized kernel for the problem.

     
  2. 2.

    For Split Edge Deletion, the problem of determining whether there are k edges whose deletion results in a split graph, we give an \({\mathcal{O}}^{*}( 2^{ {\mathcal{O}}(\sqrt{k}\log k) } )\) algorithm. We also prove the existence of an \({\mathcal{O}}(k^{2})\) kernel.

     
In addition, we note that our algorithm for Split Edge Deletion adds to the small number of subexponential parameterized algorithms not obtained through bidimensionality (Demaine et al. in J. ACM 52(6): 866–893, 2005), and on general graphs.

Keywords

Parameterized complexity Deletion problems Split graphs Subexponential algorithm 

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Esha Ghosh
    • 1
  • Sudeshna Kolay
    • 1
  • Mrinal Kumar
    • 2
  • Pranabendu Misra
    • 3
  • Fahad Panolan
    • 1
  • Ashutosh Rai
    • 1
  • M. S. Ramanujan
    • 1
  1. 1.The Institute of Mathematical SciencesChennaiIndia
  2. 2.Indian Institute of Technology, MadrasChennaiIndia
  3. 3.Chennai Mathematical InstituteChennaiIndia

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