Bulletin of the
Korean Mathematical Society BKMS

Let $\gamma_{t}(G)$ and $\tau(G)$ denote the total domination number and vertex cover number of graph $G$, respectively. In this paper, we study the total domination number of the central tree $C(T)$ for a tree $T$. First, a relationship between the total domination number of $C(T)$ and the vertex cover number of tree $T$ is discussed. We characterize the central trees with equal total domination number and independence number. Applying the first result, we improve the upper bound on the total domination number of $C(T)$ and solve one open problem posed by Kazemnejad et al..

Keywords: Total domination number, vertex cover number, central tree

MSC numbers: 05C69, 05C35

Supported by: This research was supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education (2017R1D1A3B03029912).