TY - JOUR

T1 - Quadratic vertex kernel for split vertex deletion

AU - Agrawal, Akanksha

AU - Gupta, Sushmita

AU - Jain, Pallavi

AU - Krithika, R.

N1 - Funding Information:
We are thankful to the reading course headed by Saket Saurabh, held at Institute of Mathematical Sciences, during which the problem was suggested for research. The first three authors during the work were supported by the ERC Consolidator Grant SYSTEMATIC-GRAPH (No. 725978 ) of the European Research Council ; Research Council of Norway , Toppforsk project (No. 274526 ); and SERB-NPDF fellowship ( PDF/2016/003508 ) of Department of Science and Technology , India, respectively.
Funding Information:
We are thankful to the reading course headed by Saket Saurabh, held at Institute of Mathematical Sciences, during which the problem was suggested for research. The first three authors during the work were supported by the ERC Consolidator Grant SYSTEMATIC-GRAPH (No. 725978) of the European Research Council; Research Council of Norway, Toppforsk project (No. 274526); and SERB-NPDF fellowship (PDF/2016/003508) of Department of Science and Technology, India, respectively.
Publisher Copyright:
© 2020 Elsevier B.V.

PY - 2020/9/12

Y1 - 2020/9/12

N2 - A graph is called a split graph if its vertex set can be partitioned into a clique and an independent set. Split graphs have rich mathematical structure and interesting algorithmic properties making it one of the most well-studied special graph classes. In the SPLIT VERTEX DELETION problem, given a graph and a positive integer k, the objective is to test whether there exists a subset of at most k vertices whose deletion results in a split graph. In this paper, we design a kernel for this problem with O(k2) vertices, improving upon the previous cubic bound known. Also, by giving a simple reduction from the VERTEX COVER problem, we establish that SPLIT VERTEX DELETION does not admit a kernel with O(k2−ϵ) edges, for any ϵ>0, unless NP⊆coNP/poly.

AB - A graph is called a split graph if its vertex set can be partitioned into a clique and an independent set. Split graphs have rich mathematical structure and interesting algorithmic properties making it one of the most well-studied special graph classes. In the SPLIT VERTEX DELETION problem, given a graph and a positive integer k, the objective is to test whether there exists a subset of at most k vertices whose deletion results in a split graph. In this paper, we design a kernel for this problem with O(k2) vertices, improving upon the previous cubic bound known. Also, by giving a simple reduction from the VERTEX COVER problem, we establish that SPLIT VERTEX DELETION does not admit a kernel with O(k2−ϵ) edges, for any ϵ>0, unless NP⊆coNP/poly.

KW - Kernelization

KW - Split graph

KW - Vertex deletion

UR - http://www.scopus.com/inward/record.url?scp=85085938914&partnerID=8YFLogxK

U2 - 10.1016/j.tcs.2020.06.001

DO - 10.1016/j.tcs.2020.06.001

M3 - Article

AN - SCOPUS:85085938914

SN - 0304-3975

VL - 833

SP - 164

EP - 172

JO - Theoretical Computer Science

JF - Theoretical Computer Science

ER -