On the maximum number of minimum total dominating sets in forests

TL;DR

This paper investigates upper bounds on the number of minimum total dominating sets in forests, proposing a conjecture for trees and providing bounds for general forests with no isolated vertices.

Contribution

It introduces a conjecture on the maximum number of minimum total dominating sets in trees and establishes upper bounds for forests, relaxing the conjecture.

Findings

Proposes a conjecture for trees with at least two vertices.

Provides upper bounds for forests with no isolated vertices.

Establishes multiple bounds depending on forest parameters.

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.