A Note on Inequalities for Three Domination Parameters

In this short paper, we establish relations between the domination number $\gamma$, the total domination number $\gamma_t$, and the connected domination number $\gamma_c$ of a graph. In particular, we prove upper and lower bounds for $\gamma_t$ in terms of $\gamma$ and $\gamma_c$. Moreover, we propose the following conjecture: for every connected isolated-free graph $G$, \begin{equation*}\label{eq:low} \gamma_t(G) \geq \left \lceil \frac{3\gamma(G) +2\gamma_c(G)}{6}\right\rceil. \end{equation*} As evidence to support the conjecture, we prove that the conjecture holds when $\gamma_t(G) = \gamma_c(G)$ and also, when $\gamma_t(G) = \gamma_c(G) -1$.