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arXiv · 2609.01021

Kernelization of 2-Club Cluster Edge Deletion on Interval Graphs

Ajinkya Gaikwad
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Abstract

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.

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Ajinkya Gaikwad. 2026-09-01. Kernelization of 2-Club Cluster Edge Deletion on Interval Graphs. https://arxiv.org/abs/2609.01021

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