, Volume 79, Issue 1, pp 271–290 | Cite as

Quick but Odd Growth of Cacti

  • Sudeshna Kolay
  • Daniel Lokshtanov
  • Fahad PanolanEmail author
  • Saket Saurabh


Let \({\mathcal {F}}\) be a family of graphs. Given an n-vertex input graph G and a positive integer k, testing whether G has a vertex subset S of size at most k, such that \(G-S\) belongs to \({\mathcal {F}}\), is a prototype vertex deletion problem. These type of problems have attracted a lot of attention in recent times in the domain of parameterized complexity. In this paper, we study two such problems; when \({\mathcal {F}}\) is either the family of forests of cacti or the family of forests of odd-cacti. A graph H is called a forest of cacti if every pair of cycles in H intersect on at most one vertex. Furthermore, a forest of cacti H is called a forest of odd cacti, if every cycle of H is of odd length. Let us denote by \({\mathcal {C}}\) and \({{\mathcal {C}}}_\mathsf{odd}\), the families of forests of cacti and forests of odd cacti, respectively. The vertex deletion problems corresponding to \({\mathcal {C}}\) and \({{\mathcal {C}}}_\mathsf{odd}\) are called Diamond Hitting Set and Even Cycle Transversal, respectively. In this paper we design randomized algorithms with worst case run time \(12^k n^{\mathcal {O}(1)}\) for both these problems. Our algorithms considerably improve the running time for Diamond Hitting Set and Even Cycle Transversal, compared to what is known about them.


Even Cycle Transversal Diamond Hitting Set Fixed parameter tractability Randomized algorithms 



The authors would like to thank the anonymous reviewers for their valuable comments and suggestions which helped us improve the quality of the paper.


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Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  • Sudeshna Kolay
    • 1
  • Daniel Lokshtanov
    • 2
  • Fahad Panolan
    • 2
    Email author
  • Saket Saurabh
    • 1
  1. 1.The Institute of Mathematical SciencesHBNITaramani, ChennaiIndia
  2. 2.Department of InformaticsUniversity of BergenBergenNorway

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