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In the $\ell$-Component Order Connectivity problem ($\ell \in \mathbb{N}$),we are given a graph $G$ on $n$ vertices, $m$ edges and a non-negative integer$k$ and asks whether there exists a set of vertices $S\subseteq V(G)$ such that$|S|\leq k$ and the size of the largest connected component in $G-S$ is at most$\ell$. In this paper, we give a kernel for $\ell$-Component Order Connectivitywith at most $2\ell k$ vertices that takes $n^{\mathcal{O}(\ell)}$ time forevery constant $\ell$. On the way to obtaining our kernel, we prove ageneralization of the $q$-Expansion Lemma to weighted graphs. Thisgeneralization may be of independent interest.

A $2\ell k$ Kernel for $\ell$-Component Order Connectivity