Bounds on the Exponential Domination Number

TL;DR

This paper investigates bounds on the exponential domination number in graphs, providing new lower and upper bounds that improve upon previous results, with implications for understanding domination in complex networks.

Contribution

It introduces tighter bounds on the exponential domination number, especially strengthening the upper bound for connected graphs with given parameters.

Findings

Established a new lower bound involving maximum degree and graph size.

Improved the upper bound for the exponential domination number based on graph radius.

Provided an enhanced upper bound of (43/108)(n+2) for connected graphs.

Abstract

As a natural variant of domination in graphs, Dankelmann et al. [Domination with exponential decay, Discrete Math. 309 (2009) 5877-5883] introduce exponential domination, where vertices are considered to have some dominating power that decreases exponentially with the distance, and the dominated vertices have to accumulate a sufficient amount of this power emanating from the dominating vertices. More precisely, if $S$ is a set of vertices of a graph $G$, then $S$ is an exponential dominating set of $G$ if $\sum\limits_{v\in S}\left(\frac{1}{2}\right)^{{\rm dist}_{(G,S)}(u,v)-1}\geq 1$ for every vertex $u$ in $V(G)\setminus S$, where ${\rm dist}_{(G,S)}(u,v)$ is the distance between $u\in V(G)\setminus S$ and $v\in S$ in the graph $G-(S\setminus \{ v\})$. The exponential domination number $\gamma_e(G)$ of $G$ is the minimum order of an exponential dominating set of $G$. Dankelmann et al. show $$\frac{1}{4}({\rm d}+2)\leq \gamma_e(G)\leq \frac{2}{5}(n+2)$$ for a connected graph $G$ of order $n$ and diameter ${\rm d}$. We provide further bounds and in particular strengthen their upper bound. Specifically, for a connected graph $G$ of order $n$, maximum degree $\Delta$ at least $3$, radius ${\rm r}$ at least $1$, we show \begin{eqnarray*} \gamma_e(G) & \geq & \left(\frac{n}{13(\Delta-1)^2}\right)^{\frac{\log_2(\Delta-1)+1}{\log_2^2(\Delta-1)+\log_2(\Delta-1)+1}},\\[3mm] \gamma_e(G) & \leq & 2^{2{\rm r}-2}\mbox{, and }\\[3mm] \gamma_e(G) & \leq & \frac{43}{108}(n+2). \end{eqnarray*}