Bounds on zero forcing using (upper) total domination and minimum degree

TL;DR

This paper establishes new bounds on the zero forcing number of a graph using total domination parameters, characterizes extremal graphs, and demonstrates the bounds' sharpness through infinite families.

Contribution

It introduces bounds relating zero forcing number to total domination numbers and characterizes graphs attaining these bounds, extending previous work on zero forcing parameters.

Findings

Proves bounds: Z(G)+γ_t(G) ≤ n(G) and Z(G)+Γ_t(G)/2 ≤ n(G) for graphs without isolated vertices.

Shows every graph H is an induced subgraph of a graph G where the bounds are tight.

Provides characterizations of graphs with power domination 1 and those attaining the trivial lower bound.

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 from [D.D.~Row, A technique for computing the zero forcing number of a graph with a cut-vertex, Linear Alg.\ Appl.\ 436 (2012) 4423--4432], where the graphs with zero forcing number equal to $2$ were characterized.