On anti-Kekulé and s-restricted matching preclusion problems

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