On Tight Bounds for the k-Forcing Number of a Graph

The k-forcing number of a graph G, denoted by Fk(G), was introduced by Amos et al. It is a generalization of the zero forcing number of a graph G, denoted by Z(G). Amos et al. proved that for a connected graph G of order n with maximum degree 2, Z(G , and this inequality is sharp. Moreover, they posed a conjecture that Z(G)=F1(G)= if and only if G=Cn, G=K+1 or G=K,. In this paper, we prove that this conjecture is true. Moreover, we point out a mistake in their paper and get a stronger result which shows that Fn-1(G)=1 if and only if G is connected and Fk(G)=n-k if and only if G=Kn for kn-2.