Linear Kernels for $l$-Exact Component Order Connectivity

2026-05-19

AI Summary

This study addresses the $l$-Exact Component Order Connectivity problem: given a graph $G$ and an integer $k$, determine whether at most $k$ vertices can be removed so that every remaining connected component contains exactly $l$ vertices. The authors propose a unified algorithmic framework based on extended crown decomposition, linear programming, and graph kernelization techniques, yielding a linear kernel of size $O(kl)$ for any fixed $l \geq 1$. Notably, this is the first linear-size kernel for all $l \geq 3$; for $l = 2$, the kernel size is improved from the previously known $6k$ to $3k + 1$; and for $l = 1$, the result matches the optimal kernel for Vertex Cover.

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.

Problem

• l-Exact Component Order Connectivity
• vertex deletion
• connected components
• graph kernelization
• parameterized complexity

Innovation

• linear kernel
• component order connectivity
• crown decomposition
• parameterized complexity
• vertex deletion