区间图上的2-团簇边删除问题的核化
Kernelization of 2-Club Cluster Edge Deletion on Interval Graphs
- Faculty of Information Technology, Czech Technical University in Prague（布拉格捷克理工大学信息技术学院）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文针对区间图上2-团簇边删除问题，得到大小为O(k⁵)的多项式顶点核，还证明该问题在单位区间图上多项式可解、在分裂图上NP-难，回应了其多项式核的开放问题。
AI中文摘要：
s-团簇边删除问题是指，给定图G和整数k，能否删除至多k条边，使得剩余每个连通分量的直径至多为s。该问题将经典的团簇边删除问题推广，允许分量直径有界而非要求为完全图。在一般图上，已知当参数为k时，2-团簇边删除问题是固定参数可处理的，但它是否存在多项式核仍是开放问题，如文献[ABUKHZAM2023113864]所提出。受此问题驱动，本文研究区间图上的该问题，得到了大小为O(k⁵)的多项式顶点核。作为补充结果，本文还证明s-团簇边删除问题在单位区间图上可在多项式时间内求解，且2-团簇边删除问题即使在分裂图上也是NP-难的。
英文摘要：
The \emph{$s$-Club Cluster Edge Deletion} problem asks whether, given a graph $G$ and an integer $k$, one can delete at most $k$ edges so that every remaining connected component has diameter at most~$s$. This generalizes the classical \emph{Cluster Edge Deletion} problem by permitting components of bounded diameter instead of requiring cliques. On general graphs, $2$-Club Cluster Edge Deletion is known to be fixed-parameter tractable when parameterized by $k$, but it remains open whether it admits a polynomial kernel, as posed in~\cite{ABUKHZAM2023113864}. Motivated by this question, we study the problem on interval graphs and obtain a polynomial vertex kernel of size $\mathcal{O}(k^{5})$. As a complementary result, we also show that the \emph{$s$-Club Cluster Edge Deletion} problem is polynomial time solvable on unit interval graphs. We also show that $2$-Club Cluster Edge Deletion is NP-hard even on split graphs.
