On anti-Kekulé and s -restricted matching preclusion problems

The anti-Kekulé number of a connected graph G is the smallest number of edges whose deletion results in a connected subgraph having no Kekulé structures (perfect matchings). As a common generalization of (conditional) matching preclusion number and anti-Kekulé 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-Kekulé 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.