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

如果可以将图的顶点集划分为集团和独立集，则将其称为拆分图。拆分图具有丰富的数学结构和有趣的算法属性，使其成为研究最深入的特殊图类之一。在拆分顶点删除问题中，给定一个图和一个正整数k，目的是测试是否存在最多k个顶点的子集，其删除会导致一个拆分图。在本文中，我们针对此问题设计了一个内核$\mathcal{Ø}（{ķ}^2）$顶点，在已知的先前三次边界的基础上进行了改进。另外，通过简单地减少“顶点覆盖”问题，我们确定“拆分顶点删除”不接受带有以下内容的内核：$\mathcal{Ø}（{ķ}^{2-ϵ}）$ 边缘，任何 $ϵ>0$，除非 $\mathsf{\text{NP}}\subseteq \mathsf{\text{coNP / poly}}$。