边修改问题的改进核
Improved Kernels for Edge Modification Problems
摘要：
在边修改问题中，我们被要求至多修改给定图的$k$边，以使图满足一定的性质。根据允许的操作，我们有完成问题和边删除问题。大量的努力已经投入到理解这些问题的内核化的复杂性。我们重新审视了几个研究得很好的边修改问题，并为它们开发了改进的内核：\begin{itemize} \item a $2 k$-vertex kernel for the cluster edge deletion problem, \item a $3 k^2$-vertex kernel for the trivially perfect completion problem, \item a $5 k^{1.5}$-vertex kernel for the split completion problem and the split edge deletion problem, and \item a $5 k^{1.5}$-vertex kernel for the pseudo-split completion problem and the pseudo-split edge deletion problem. \end{itemize}此外，我们的分裂完成和伪分裂完成的内核只有$O(k^{2.5})$边缘。我们的结果还包括一个$2 k$ -顶点核的强三元闭包问题，这是有关集群边缘删除。
英文摘要：
In an edge modification problem, we are asked to modify at most $k$ edges to a given graph to make the graph satisfy a certain property. Depending on the operations allowed, we have the completion problems and the edge deletion problems. A great amount of efforts have been devoted to understanding the kernelization complexity of these problems. We revisit several well-studied edge modification problems, and develop improved kernels for them: \begin{itemize} \item a $2 k$-vertex kernel for the cluster edge deletion problem, \item a $3 k^2$-vertex kernel for the trivially perfect completion problem, \item a $5 k^{1.5}$-vertex kernel for the split completion problem and the split edge deletion problem, and \item a $5 k^{1.5}$-vertex kernel for the pseudo-split completion problem and the pseudo-split edge deletion problem. \end{itemize} Moreover, our kernels for split completion and pseudo-split completion have only $O(k^{2.5})$ edges. Our results also include a $2 k$-vertex kernel for the strong triadic closure problem, which is related to cluster edge deletion.
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