A Note on Inequalities for Three Domination Parameters

TL;DR

This paper explores relationships between three domination parameters in graphs, establishing bounds and proposing a conjecture supported by proofs in specific cases, advancing understanding of domination inequalities.

Contribution

It introduces new bounds for total domination in terms of domination and connected domination, and proposes a conjecture linking these parameters with partial proofs.

Findings

Established bounds for mma_t in terms of mma and mma_c.

Proposed a conjecture relating mma_t, mma, and mma_c.

Proved the conjecture holds when mma_t = mma_c or mma_t = mma_c - 1.

Abstract

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 \lfloor \frac{3\gamma(G) +2\gamma_c(G)}{6}\right\rfloor. \end{equation*} As evidence to support the conjecture, we prove that the conjecture holds when $\gamma_t(G) = \gamma_c(G)$ and also, when $\gamma_t(G) = \gamma_c(G) -1$.