Over the past decades, various metrics have emerged in graph theory to grasp the complex nature of network vulnerability. In this paper, we study two specific measures: (weighted) vertex integrity (wVI) and (weighted) component order connectivity (wCOC). These measures not only evaluate the number of vertices required to decompose a graph into fragments, but also take into account the size of the largest remaining component. The main focus of our paper is on kernelization algorithms tailored to both measures. We capitalize on the structural attributes inherent in different crown decompositions, strategically combining them to introduce novel kernelization algorithms that advance the current state of the field. In particular, we extend the scope of the balanced crown decomposition provided by Casel et al.~[7] and expand the applicability of crown decomposition techniques. In summary, we improve the vertex kernel of VI from $p^3$ to $p^2$, and of wVI from $p^3$ to $3(p^2 + p^{1.5} p_{\ell})$, where $p_{\ell} < p$ represents the weight of the heaviest component after removing a solution. For wCOC we improve the vertex kernel from $\mathcal{O}(k^2W + kW^2)$ to $3\mu(k + \sqrt{\mu}W)$, where $\mu = \max(k,W)$. We also give a combinatorial algorithm that provides a $2kW$ vertex kernel in FPT-runtime when parameterized by $r$, where $r \leq k$ is the size of a maximum $(W+1)$-packing. We further show that the algorithm computing the $2kW$ vertex kernel for COC can be transformed into a polynomial algorithm for two special cases, namely when $W=1$, which corresponds to the well-known vertex cover problem, and for claw-free graphs. In particular, we show a new way to obtain a $2k$ vertex kernel (or to obtain a 2-approximation) for the vertex cover problem by only using crown structures.

翻译：在过去的几十年中，图论领域涌现了多种度量指标，以揭示网络脆弱性的复杂本质。本文研究两种特定度量：(加权)顶点完整性(wVI)和(加权)分量阶连通性(wCOC)。这些度量不仅评估将图分解为碎片所需的顶点数量，还考虑了剩余最大连通分量的大小。本文主要关注针对这两种度量的核化算法。我们利用不同冠分解中固有的结构属性，通过策略性组合这些结构，提出了一系列推进领域前沿的新型核化算法。具体而言，我们扩展了Casel等人[7]提出的平衡冠分解的应用范围，并拓宽了冠分解技术的适用性。总结而言：我们将VI的顶点核从$p^3$降至$p^2$，将wVI的顶点核从$p^3$降至$3(p^2 + p^{1.5} p_{\ell})$，其中$p_{\ell} < p$表示移除解后最重分量的权重；针对wCOC，我们将顶点核从$\mathcal{O}(k^2W + kW^2)$优化至$3\mu(k + \sqrt{\mu}W)$，其中$\mu = \max(k,W)$。此外，我们提出一种组合算法，在参数$r$（$r \leq k$为最大$(W+1)$-打包的大小）约束的FPT运行时间内得到$2kW$顶点核。进一步证明，COC的$2kW$顶点核算法可转化为两个特例的多项式算法：当$W=1$时对应著名的顶点覆盖问题，以及针对无爪图的场景。特别地，我们展示了一种仅通过冠结构获得顶点覆盖问题的$2k$顶点核（或2-近似解）的新方法。