Quadratic Vertex Kernel for Split Vertex Deletion

TLDR: A kernel for the Split Vertex Deletion (SVD) problem is designed, improving upon the previous cubic bound known and establishing that SVD does not admit a kernel with \(\mathcal {O}(k^{2-\epsilon })\) edges, for any \(\ep silon >0\), unless \(\textsf {NP}\subseteq \textsf{coNP/poly}\).

Abstract: 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(SVD) 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 \(\mathcal {O}(k^2)\) vertices, improving upon the previous cubic bound known. Also, by giving a simple reduction from the Vertex Cover problem, we establish that SVD does not admit a kernel with \(\mathcal {O}(k^{2-\epsilon })\) edges, for any \(\epsilon >0\), unless \(\textsf {NP}\subseteq \textsf {coNP/poly}\).