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Mathematics
Given a graph G, the zero forcing number of G, Z(G), is the minimum cardinality of any set S of v... more 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 eith...
Mathematics
Given a graph G, the zero forcing number of G, Z(G), is the minimum cardinality of any set S of v... more 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 eith...