Vertex integrity is a graph parameter that measures the connectivity of a graph. Informally, its meaning is that a graph has small vertex integrity if it has a small separator whose removal disconnects the graph into connected components which are themselves also small. Graphs with low vertex integrity are extremely structured; this renders many hard problems tractable and has recently attracted interest in this notion from the parameterized complexity community. In this paper we revisit the NP-complete problem of computing the vertex integrity of a given graph from the point of view of structural parameterizations. We present a number of new results, which also answer some recently posed open questions from the literature. Specifically: We show that unweighted vertex integrity is W[1]-hard parameterized by treedepth; we show that the problem remains W[1]-hard if we parameterize by feedback edge set size (via a reduction from a Bin Packing variant which may be of independent interest); and complementing this we show that the problem is FPT by max-leaf number. Furthermore, for weighted vertex integrity, we show that the problem admits a single-exponential FPT algorithm parameterized by vertex cover or by modular width, the latter result improving upon a previous algorithm which required weights to be polynomially bounded.

翻译：顶点完整性是衡量图连通性的图参数。非正式地，其含义是若图存在一个小的分离子，去除该分离子后图被划分为同样较小的连通分量，则该图具有较小的顶点完整性。低顶点完整性的图具有高度结构性；这使得许多困难问题变得易处理，并最近引起了参数化复杂性学界对这一概念的关注。本文从结构性参数化的视角重新审视计算给定图顶点完整性这一NP完全问题。我们提出了一系列新结果，其中部分还回答了文献中最近提出的开放问题。具体而言：我们证明无权顶点完整性关于树深参数化是W[1]-难的；我们证明该问题关于反馈边集大小参数化仍是W[1]-难的（通过一个可能具有独立意义的装箱问题变体的归约）；与此互补，我们证明该问题关于最大叶子数参数化是FPT的。此外，对于加权顶点完整性，我们证明该问题关于顶点覆盖或模宽度存在单指数FPT算法，后者改进了需要权重多项式有界的先前算法。