On anti-Kekul\'{e} and $s$-restricted matching preclusion problems

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.