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

Let $F(G)$, Fₜ(G), β(G), and β'(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) = (Δ-2) β(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) ≤ β'(T)-1 and Fₜ(T) ≤ β'(T) for any tree T of order n ≥ 3. Moreover, we also characterize all trees T with F(T) = β'(T)-1 and F(T) = β'(T)-2, respectively.