Leaky Forcing: A New Variation of Zero Forcing

Abstract: Zero forcing is a one-player game played on a graph. The player chooses some set of vertices to color, then iteratively applies a color change rule: If all but one of a colored vertex's neighbors are colored, color (i.e. "force") the remaining uncolored neighbor. Generally, the goal is to find the minimum number of vertices to initially color such that all vertices eventually become colored. Recently, equivalent formations of zero forcing have been developed in different settings including sensor allocation to solve linear systems (K.-Lin 2018), controllability in follower-leader dynamics (Monshizadeh-Zhang-Camlibel 2014), and edge covering in specific hypergraphs (Brimkov-Fast-Hicks 2016). While many variations of zero forcing are motivated by an associated minimum rank problem, these new formulations give new inspiration for new meaningful zero forcing variants. In our case, we study a new variation based on the linear algebraic interpretation mentioned above. In particular, what if there is a juncture in a network that has a leak, and, hence, is unreliable to facilitate solving a linear system on the network? In the context of zero forcing this corresponds to the following variation we call $\ell$-forcing: Given $\ell$, find a set of vertices such that for any set of $\ell$ vertices that are unable to force, all vertices will still be colored. We compute the $\ell$-forcing number for selected families of graphs including grid graphs. Perhaps surprisingly, we find examples where additional edges make the graph more "resilient" to these leaks. Further, we also implement known computational methods for our new leaky forcing variation.