On the maximum number of minimum total dominating sets in forests

Abstract

We propose the conjecture that every tree with order $n$ at least $2$ and total domination number $\gamma_t$ has at most $\left(\frac{n-\frac{\gamma_t}{2}}{\frac{\gamma_t}{2}}\right)^{\frac{\gamma_t}{2}}$ minimum total dominating sets. As a relaxation of this conjecture, we show that every forest $F$ with order $n$, no isolated vertex, and total domination number $\gamma_t$ has at most $\min\left\{\left(8\sqrt{e}\, \right)^{\gamma_t}\left(\frac{n-\frac{\gamma_t}{2}}{\frac{\gamma_t}{2}}\right)^{\frac{\gamma_t}{2}}, (1+\sqrt{2})^{n-\gamma_t},1.4865^n\right\}$ minimum total dominating sets.