Blocking zero forcing processes in Cartesian products of graphs

Abstract

In a zero forcing process, an initial vertex coloring of a graph is updated iteratively according to the following conversion rule: an uncolored vertex becomes colored if it is the only uncolored neighbor of some colored vertex. In this process, a zero forcing set is an initial set of colored vertices such that all the remaining vertices will ultimately change from uncolored to colored, and the zero forcing number is the minimum cardinality of a zero forcing set. We investigate two related concepts: a zero blocking set is the complement of a set which is not a zero forcing set, and the zero blocking number is the minimum cardinality of a zero blocking set. We provide a tight upper bound for the zero blocking number of Cartesian products of two arbitrary graphs. Exact values that are smaller than this general upper bound are also established when one of the graphs in the product is a cycle and the other is a path with at least as many vertices as the cycle.