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.

## Keywords

Vertex Cover Reduction Rule Graph Class Split Graph Colored Instance
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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© Springer-Verlag Berlin Heidelberg 2012

## Authors and Affiliations

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