SWAT 2012: Algorithm Theory – SWAT 2012 pp 107-118

# Faster Parameterized Algorithms for Deletion to Split Graphs

• Esha Ghosh
• Sudeshna Kolay
• Mrinal Kumar
• Pranabendu Misra
• Ashutosh Rai
• M. S. Ramanujan
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7357)

## 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. 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. 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.

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## Authors and Affiliations

• Esha Ghosh
• 1
• Sudeshna Kolay
• 1
• Mrinal Kumar
• 2
• Pranabendu Misra
• 3