On leaky forcing and resilience

A leak is a vertex that is not allowed to perform a force during the zero forcing process. Leaky forcing was recently introduced as a new variation of zero forcing in order to analyze how leaks in a network disrupt the zero forcing process. The -leaky forcing number of a graph is the size of the smallest zero forcing set that can force a graph despite leaks. A graph is -resilient if its zero forcing number is the same as its -leaky forcing number. In this paper, we analyze -leaky forcing and show that if an -leaky forcing set is robust enough, then is an -leaky forcing set. This provides the framework for characterizing -leaky forcing sets. Furthermore, we consider structural implications of -resilient graphs. We apply these results to bound the -leaky forcing number of several graph families including trees, supertriangles, and grid graphs. In particular, we resolve a question posed by Dillman and Kenter concerning the upper bound on the 1-leaky forcing number of grid graphs.