We study how to sparsify connectivity in graphs under a tight deletion budget. Given a graph $G$ and integers $k,x \ge 0$, Critical Node Cut (CNC) asks whether we can delete at most $k$ vertices so that the number of remaining unordered pairs of connected vertices is at most $x$. CNC generalizes Vertex Cover (the case $x=0$) and models tasks in network design, epidemiology, and social network analysis. We comprehensively map the structural parameterized complexity landscape for Critical Node Cut. First, we prove W[1]-hardness for the combined parameter $k + \mathrm{fes} + Δ+ \mathrm{pw}$, where $\mathrm{fes}$ is the feedback edge set number, $Δ$ the maximum degree, and $\mathrm{pw}$ the pathwidth of the input graph respectively. This significantly improves over the known W[1]-hardness for $k+\mathrm{tw}$, where $\mathrm{tw}$ denotes the treewidth, and is tight in that tree-depth together with maximum degree trivially yields FPT. Second, we give new positive results. Specifically, we identify three structural parameters--max-leaf number, vertex integrity, and modular-width--that render the problem fixed-parameter tractable, and develop a polynomial-time algorithm for graphs of constant clique-width. Third, leveraging a technique introduced by Lampis~[ICALP '14], we develop an FPT approximation scheme that, for any $\varepsilon > 0$, computes a $(1+\varepsilon)$-approximate solution in time $(\mathrm{tw} / \varepsilon)^{\mathcal{O}(\mathrm{tw})} n^{\mathcal{O}(1)}$, where $\mathrm{tw}$ denotes the treewidth of the input graph. Finally, we show that CNC admits no polynomial kernel when parameterized by vertex cover number, unless standard assumptions fail. Together, these results substantially sharpen the known complexity landscape for CNC.

翻译：本研究探讨如何在严格删除预算下稀疏化图的连通性。给定图$G$与整数$k,x \ge 0$，关键节点割（CNC）问题要求判断是否存在至多$k$个顶点的删除方案，使得剩余连通顶点无序对的数量不超过$x$。CNC问题推广了顶点覆盖问题（$x=0$的情形），并为网络设计、流行病学及社交网络分析中的任务提供建模框架。本文系统刻画了关键节点割问题的结构参数化复杂度全景。首先，我们证明了该问题对于组合参数$k + \mathrm{fes} + Δ+ \mathrm{pw}$是W[1]-难的，其中$\mathrm{fes}$表示反馈边集数，$Δ$为最大度数，$\mathrm{pw}$为输入图的路径宽度。这一结论显著改进了已知的$k+\mathrm{tw}$参数下的W[1]-难性结果（$\mathrm{tw}$表示树宽），且具有紧致性：结合树深与最大度数的参数可平凡地得到固定参数可解性。其次，我们提出了新的积极结果。具体而言，我们识别出三个能使问题转化为固定参数可解的结构参数——最大叶数、顶点完整度与模宽，并针对恒定团宽图设计了多项式时间算法。第三，基于Lampis~[ICALP '14]提出的技术，我们开发了固定参数近似方案：对于任意$\varepsilon > 0$，该方案可在$(\mathrm{tw} / \varepsilon)^{\mathcal{O}(\mathrm{tw})} n^{\mathcal{O}(1)}$时间内计算$(1+\varepsilon)$-近似解，其中$\mathrm{tw}$表示输入图的树宽。最后，我们证明当以顶点覆盖数作为参数时，除非标准复杂性假设不成立，否则CNC问题不存在多项式核。这些结果共同显著深化了CNC问题的已知复杂度认知。