The anti-Kekul\'{e} number of a connected graph $G$ is the smallest number of edges whose deletion results in a connected subgraph having no Kekul\'{e} structures (perfect matchings). As a common generalization of (conditional) matching preclusion number and anti-Kekul\'{e} number of a graph $G$, we introduce $s$-restricted matching preclusion number of $G$ as the smallest number of edges whose deletion results in a subgraph without perfect matchings such that each component has at least $s+1$ vertices. In this paper, we first show that conditional matching preclusion problem and anti-Kekul\'{e} problem are NP-complete, respectively, then generalize this result to $s$-restricted matching preclusion problem. Moreover, we give some sufficient conditions to compute $s$-restricted matching preclusion numbers of regular graphs. As applications, $s$-restricted matching preclusion numbers of complete graphs, hypercubes and hyper Petersen networks are determined.

翻译：连通图$G$的反Kekulé数是指删除最少数量的边后，所得连通子图不再含有Kekulé结构（完美匹配）的边数。作为图$G$的（条件性）匹配排除数与反Kekulé数的共同推广，我们引入$G$的$s$-受限匹配排除数，定义为删除最少数量的边后，所得子图无完美匹配且每个连通分支至少包含$s+1$个顶点。本文首先分别证明条件性匹配排除问题和反Kekulé问题是NP完全的，进而将此结论推广至$s$-受限匹配排除问题。此外，我们给出若干充分条件用于计算正则图的$s$-受限匹配排除数。作为应用，确定了完全图、超立方体图及超Petersen网络的$s$-受限匹配排除数。