The connectivity of a graph is an important parameter to evaluate its reliability. $k$-restricted connectivity (resp. $R^h$-restricted connectivity) of a graph $G$ is the minimum cardinality of a set $S$ of vertices in $G$, if exists, whose deletion disconnects $G$ and leaves each component of $G-S$ with more than $k$ vertices (resp. $δ(G-S)\geq h$). In contrast, structure (substructure) connectivity of $G$ is defined as the minimum number of vertex-disjoint subgraphs whose deletion disconnects $G$. As generalizations of the concept of connectivity, structure (substructure) connectivity, restricted connectivity and $R^h$-restricted connectivity have been extensively studied from the combinatorial point of view. Very little is known about the computational complexity of these variants, except for the recently established NP-completeness of $k$-restricted edge-connectivity. In this paper, we prove that the problems of determining structure, substructure, restricted, and $R^h$-restricted connectivity are all NP-complete.

翻译：图的连通度是评估其可靠性的重要参数。图$G$的$k$-限制连通度（相应地，$R^h$-限制连通度）定义为：若存在，使得删除后$G$不连通，且$G-S$的每个分支包含多于$k$个顶点（相应地，满足$δ(G-S)\geq h$）的顶点集$S$的最小基数。相对地，图$G$的结构（子结构）连通度则定义为：使得删除后$G$不连通的顶点不相交子图的最小数目。作为连通度概念的推广，结构（子结构）连通度、限制连通度及$R^h$-限制连通度已从组合角度被广泛研究。然而，除最近确立的$k$-限制边连通度的NP完全性外，这些变体的计算复杂性知之甚少。本文证明，判定结构连通度、子结构连通度、限制连通度及$R^h$-限制连通度的问题均为NP完全问题。