The Zero (Total) Forcing Number and Covering Number of Trees

Let F(G), Ft(G), beta(G), and beta '(G) be the zero forcing number, the total forcing number, the vertex covering number and the edge covering number of a graph G, respectively. In this paper, we first completely characterize all trees T with F(T)=(Delta-2)beta(T)+1, solving a problem proposed by Brimkov et al. in 2023. Next, we study the relationship between the zero (or total) forcing number of a tree and its edge covering number, and show that F(T)<=beta '(T)-1and Ft(T)<=beta '(T)for any tree T of order n >= 3. Moreover, we also characterize all trees T with F(T)=beta '(T)-1 and F(T)=beta '(T)-2, respectively.