Geodesic
On the maximum number of minimum total dominating sets in forests
Discrete mathematics & theoretical computer science, Tome 21 (2019) no. 3

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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. 
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     author = {Henning, Michael A. and Mohr, Elena and Rautenbach, Dieter},
     title = {On the maximum number of minimum total dominating sets in forests},
     journal = {Discrete mathematics & theoretical computer science},
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Henning, Michael A.; Mohr, Elena; Rautenbach, Dieter. On the maximum number of minimum total dominating sets in forests. Discrete mathematics & theoretical computer science, Tome 21 (2019) no. 3. doi: 10.23638/DMTCS-21-3-3

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