Characterization of All Graphs with a Failed Skew Zero Forcing Number of 1

Given a graph G , the zero forcing number of G , Z ( G ) , is the minimum cardinality of any set S of vertices of which repeated applications of the forcing rule results in all vertices being in S . The forcing rule is: if a vertex v is in S , and exactly one neighbor u of v is not in S , then u is added to S in the next iteration. Hence the failed zero forcing number of a graph was defined to be the cardinality of the largest set of vertices which fails to force all vertices in the graph. A similar property called skew zero forcing was defined so that if there is exactly one neighbor u of v is not in S , then u is added to S in the next iteration. The difference is that vertices that are not in S can force other vertices. This leads to the failed skew zero forcing number of a graph, which is denoted by F − ( G ) . In this paper, we provide a complete characterization of all graphs with F − ( G ) = 1 . Fetcie, Jacob, and Saavedra showed that the only graphs with a failed zero forcing number of 1 are either: the union of two isolated vertices; P 3 ; K 3 ; or K 4 . In this paper, we provide a surprising result: changing the forcing rule to a skew-forcing rule results in an infinite number of graphs with F − ( G ) = 1 .