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 γ_t(G) (resp. Γ_t(G)) of G. We prove that Z(G)+γ_t(G)≤n(G) and Z(G)+Γ_t(G)2≤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)+Γ_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)≥δ(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.