Zero forcing of generalized hierarchical products

Abstract

Zero forcing is a process that passes information along a network. In particular, a vertex that has information may pass that information to one of its neighbors only if that neighbor is its only neighbor that does not have that information. If the information is given to an initial set of vertices that eventually passes the information to the entire network, then that initial set is called a zero forcing set. The zero forcing number of a graph is the size of a minimum zero forcing set of that graph. In this paper, bounds on the zero forcing number of generalized hierarchical products, which are a generalization of the Cartesian product, are provided. In particular, the zero forcing number is determined exactly for certain hierarchical products by constructing zero forcing sets in the product that utilize knowledge about forcing chains for each factor of the product.