Linear Kernels for $l$-Exact Component Order Connectivity

TL;DR

This paper introduces a linear kernelization algorithm for the $l$-Exact Component Order Connectivity problem, improving kernel sizes for specific cases and utilizing crown decompositions and linear programming.

Contribution

It provides the first linear kernel for fixed $l \\geq 3$ and improves kernel bounds for $l=2$, using novel crown decomposition techniques.

Findings

Achieves an $O(kl)$-vertex kernel for the problem.

Matches the best-known kernel size for $l=1$ (Vertex Cover).

Improves kernel size for $l=2$ to $(3k+1)$ vertices.

Abstract

The \textsc{$l$-Exact Component Order Connectivity} problem asks whether, given an input graph $G$ and an integer $k$, there exists a vertex subset $S\subseteq V(G)$ of size at most $k$ such that every connected component in $G - S$ has exactly $l$ vertices. In this paper, we present an $O(kl)$-vertex kernel for this problem, computable in $|V(G)|^{O(l)}$ time. This is the first known linear kernel for each fixed $l\geq 3$. For $l=1$, this problem reduces to the classical \textsc{Vertex Cover}, and our result matches the best-known $2k$-vertex kernel. For $l=2$ (known as \textsc{Deletion to Induced Matching}), we can get a $(3k + 1)$-vertex kernel, improving the previously known result of $6k$ vertices. Our kernelization algorithm is built upon on an extended crown decomposition combined with linear programming and other techniques.