Bound for the k-Fault-Tolerant Power-Domination Number

A set S⊆V is referred to as a k-fault-tolerant power-dominating set of a given graph G=(V,E) if the difference S∖F remains a power-dominating set of G for any F⊆S with |F|≤k, where k is an integer with 0≤k<|V|. The lowest cardinality of a k-fault-tolerant power-dominating set is the k-fault-tolerant power-domination number of G, denoted by γPk(G). Generalized Petersen graphs GP(m,k) and generalized cylinders SG are two well-known graph classes. In this paper, we calculate the k-fault-tolerant power-domination number of the generalized Petersen graphs GP(m,1) and GP(m,2). Also, we obtain γPk(G) for the subclasses of cylinders SCm and SBm.