Bounds on Zero Forcing Using (Upper) Total Domination and Minimum Degree

While a number of bounds are known on the zero forcing number Z(G) of a graph G expressed in terms of the order of a graph and maximum or minimum degree, we present two bounds that are related to the (upper) total domination number \(\gamma _t(G)\) ${\gamma }_t(G)$ (resp.  \(\Gamma _t(G)\) ${\mathrm{\Gamma }}_t(G)$ ) of G. We prove that \(Z(G)+\gamma _t(G)\le n(G)\) $Z(G)+{\gamma }_t(G)\le n(G)$ and \(Z(G)+\frac{\Gamma _t(G)}{2}\le n(G)\) $Z(G)+\frac{{\mathrm{\Gamma }}_t(G)}{2}\le n(G)$ holds for any graph G with no isolated vertices of order n(G). Both bounds are sharp as demonstrated by several infinite families of graphs. In particular, we show that every graph H is an induced subgraph of a graph G with \(Z(G)+\frac{\Gamma _t(G)}{2}=n(G)\) $Z(G)+\frac{{\mathrm{\Gamma }}_t(G)}{2}=n(G)$ . Furthermore, we prove a characterization of graphs with power domination equal to 1, from which we derive a characterization of the extremal graphs attaining the trivial lower bound \(Z(G)\ge \delta (G)\) $Z(G)\ge \delta (G)$ . The class of graphs that appears in the corresponding characterizations is obtained by extending an idea of Row for characterizing the graphs with zero forcing number equal to 2.