On extremal graphs for zero forcing number (Q2102756)

For a graph \(G\) with \(S\subseteq V(G)\), \(S\) is a zero forcing set of \(G\) if iteratively adding vertices to \(S\) from \(V(G)\setminus S\) that are the unique neighbor in \(V(G)\setminus S\) of some vertex in \(S\), results in the entire \(V(G)\) of \(G\). The zero (resp., connected) forcing number, denoted by \(F(G)\) (resp., \(F_c(G)\)), of a graph \(G\) is the minimum cardinality of a zero (resp., connected) forcing set of \(G\). In this paper, some upper bounds on \(F(G)\) and \(F_c(G)\) are obtained in terms of the order and the girth of a graph \(G\), and the corresponding extremal graphs are also characterized.