Leaky forcing: a new variation of zero forcing

Zero forcing is a one -player game played on a graph. The player chooses some set of vertices to color blue, then iteratively applies a color -change rule, allowing vertices to "force" their neighbors to become blue. The results of the game determine linear -algebraic properties of matrices with a corresponding sparsity pattern. In this article, we introduce and study a new variation of zero forcing where l >= 0 of the vertices may have a "leak" which cannot facilitate any forces. The key is that the locations of the leaks are unknown at the start of the game; hence, to win, the player must implement a strategy that overcomes any configuration of l leaks. As such, this variation of zero forcing corresponds to resiliency in solving linear systems. We compute the l-leaky forcing numbers for selected families of graphs for various values of l, including grid graphs and hypercubes. We find examples where additional edges make the graph more "resilient" to these leaks. Finally, we adapt known computational methods to our new leaky forcing variation.