No abstract

Given a graph G = (V, E) and a set of vertices marked as filled, we consider a color-change rule known as zero forcing. A set S is a zero forcing set if filling S and applying all possible instances of the color change rule causes all vertices in V to be filled. A failed zero forcing set is a set of vertices that is not a zero forcing set. Given a graph G, the failed zero forcing number F (G) is the maximum size of a failed zero forcing set. In [2], the authors asked whether given any k there is a an ℓ such that all graphs with at least ℓ vertices must satisfy F (G) ≥ k. We answer this question affirmatively by proving that for a graph G with n vertices, F (G) ≥ ⌊ n−1 2 ⌋.