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

In the `-Component Order Connectivity problem (` ∈ 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 ⊆ V (G) such that |S| ≤ k and the size of the largest connected component in G−S is at most `. In this paper, we give a kernel for `-Component Order Connectivity with at most 2`k vertices that takes nO(`) time for every constant `. On the way to obtaining our kernel, we prove a generalization of the q-Expansion Lemma to weighted graphs. This generalization may be of independent interest. 1998 ACM Subject Classification F.2.2 Nonnumerical Algorithms and Problems