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A Note on Inequalities for Three Domination Parameters

Dickson Y. B. Annor

Abstract

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 $γ_t(G) = γ_c(G) -1$.

A Note on Inequalities for Three Domination Parameters

Abstract

In this short paper, we establish relations between the domination number , the total domination number , and the connected domination number of a graph. In particular, we prove upper and lower bounds for in terms of and . Moreover, we propose the following conjecture: for every connected isolated-free graph , \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 and also, when .

Paper Structure

This paper contains 3 sections, 16 theorems, 15 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Main results
  3. Conclusion

Key Result

Theorem 1

If $G$ is an isolated-free graph, then $\gamma(G) \leqslant \gamma_t(G) \leqslant 2\gamma(G)$.

Figures (3)

  • Figure 1: Graph $H$.
  • Figure 2: Graph $G'$.
  • Figure 3: A tree $T$ with $\gamma(T)=\gamma_t(T)=6$ and $\gamma_c(T)=10$.

Theorems & Definitions (26)

  • Theorem 1: bollobas1979graph
  • Theorem 2: sampathkumar1979connected
  • Theorem 3
  • proof
  • Theorem 4
  • Theorem 5
  • proof
  • Proposition 6
  • Remark 1
  • Conjecture 7
  • ...and 16 more