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：顶点完整性是一个衡量图连通性的图参数。非正式地说，若一个图具有较小的分离集，移除该分离集后图分解为同样较小的连通分量，则称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化特征，这使许多困难问题变得可处理，因此近年来引起了参数化复杂度领域对该概念的关注。本文从结构参数化角度重新审视计算给定图顶点完整性这一NP完全问题。我们提出若干新结果，同时回答了文献中近期提出的一些开放问题。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；通过一个可能具有独立兴趣的装箱问题变种的归约，我们证明若以反馈边集大小为参数化参数，该问题仍保持W[1]-难；作为补充，我们证明该问题关于最大叶子数参数化是固定参数可解的。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参数化存在单指数固定参数可解算法，后者结果改进了先前要求权重多项式有界的算法。