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

Abstract

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 \(\gamma _t(G)\) (resp. \(\Gamma _t(G)\)) of G. We prove that \(Z(G)+\gamma _t(G)\le n(G)\) and \(Z(G)+\frac{\Gamma _t(G)}{2}\le n(G)\) 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 \(Z(G)+\frac{\Gamma _t(G)}{2}=n(G)\). 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 \(Z(G)\ge \delta (G)\). 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.