Abstract
Let $F(G)$, $F_{t}(G)$, $\beta(G)$, 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 $F(T) \leq \beta'(T)-1$ and $F_{t}(T) \leq \beta'(T)$ for any tree $T$ of order $n \geq 3$. Moreover, we also characterize all trees $T$ with $F(T) = \beta'(T)-1$ and $F(T) = \beta'(T)-2$, respectively.
Funding Statement
This work was partially supported by NSFC (Nos. 12171089, 12271235), NSF of Fujian (No. 2021J02048).
Acknowledgments
The authors would like to thank the referees for their constructive corrections and valuable comments, which have considerably improved the presentation of this paper.
Citation
Dongxin Tu. Jianxi Li. Wai-Chee Shiu. "The Zero (Total) Forcing Number and Covering Number of Trees." Taiwanese J. Math. 28 (4) 657 - 670, August, 2024. https://doi.org/10.11650/tjm/240305
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