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| Main Author: | |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2506.03646 |
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Table of Contents:
- In this short paper, we establish relations between the domination number $γ$, the total domination number $γ_t$, and the connected domination number $γ_c$ of a graph. In particular, we prove upper and lower bounds for $γ_t$ in terms of $γ$ and $γ_c$. Moreover, we propose the following conjecture: for every connected isolated-free graph $G$, \begin{equation*}\label{eq:low} γ_t(G) \geq \left \lfloor \frac{3γ(G) +2γ_c(G)}{6}\right\rfloor. \end{equation*} As evidence to support the conjecture, we prove that the conjecture holds when $γ_t(G) = γ_c(G)$ and also, when $γ_t(G) = γ_c(G) -1$.