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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 ⌋.

We study the relationship between algebraic structures and their inverse semigroups of partial automorphisms. We consider a variety of classes of natural structures including equivalence structures, orderings, Boolean algebras, and relatively complemented distributive lattices. For certain subsemigroups of these inverse semigroups, isomorphism (elementary equivalence) of the subsemigroups yields isomorphism (elementary equivalence) of the underlying structures. We also prove that for some classes of computable structures, we can reconstruct a computable structure, up to computable isomorphism, from the isomorphism type of its inverse semigroup of computable partial automorphisms.

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