Faster parameterized algorithms for deletion to split graphs

  • Authors:

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

  • Affiliations:

  • The Institute of Mathematical Sciences, Chennai, India;The Institute of Mathematical Sciences, Chennai, India;Indian Institute of Technology, Madras, India;Chennai Mathematical Institute, India;The Institute of Mathematical Sciences, Chennai, India;The Institute of Mathematical Sciences, Chennai, India;The Institute of Mathematical Sciences, Chennai, India

  • Venue:
  • SWAT'12 Proceedings of the 13th Scandinavian conference on Algorithm Theory
  • Year:
  • 2012

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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 study the problem of deleting the minimum number of vertices or edges from a given input graph so that the resulting graph is split.We initiate a systematic study and give efficient fixed-parameter algorithms and polynomial sized kernels for the problem. More precisely, 1 for Split Vertex Deletion, the problem of determining whether there are k vertices whose deletion results in a split graph, we give an ${\cal O}^*(2^k)$ algorithm improving on the previous best bound of ${\cal O}^*({2.32^k})$. We also give an ${\cal O}(k^3)$-sized kernel for the problem. 2 For Split Edge Deletion, the problem of determining whether there are k edges whose deletion results in a split graph, we give an ${\cal O}^*( 2^{ O(\sqrt{k}\log k) })$ algorithm. We also prove the existence of an ${\cal O}(k^2)$ 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, and on general graphs.