Jacob, Bonnie C. "Graphs in which all maximal failed zero forcing sets are maximum." Joint Mathematics Meetings. American Mathematical Society. Seattle, WA. 11 Jan. 2025. Conference Presentation. *
A failed zero forcing set on a graph G is simply a set of vertices that do not form a zero forcing set of G. The maximum cardinality of any failed zero forcing set on is known as the failed zero forcing number of G and is denoted F(G). For many graphs, there exist maximal failed zero forcing sets with fewer than F(G) vertices. However, in some graphs, every maximal failed zero forcing set is also a maximum failed zero forcing set. We call such graphs well-failed graphs. In this talk, we establish some preliminary results about well-failed graphs, and discuss the relationship with forts as well.