Lower Bounds for the Exponential Domination Number of $C_m \times C_n$

TL;DR

This paper investigates the porous exponential domination number of grid graphs formed by Cartesian products of cycles, providing improved lower bounds using linear programming techniques.

Contribution

It introduces a linear programming approach to refine the lower bounds for the exponential domination number of $C_m imes C_n$, surpassing previous bounds.

Findings

Established a new lower bound of $rac{mn}{13.7619 + ext{small error}}$

Improved understanding of exponential domination in grid graphs

Supports conjecture on asymptotic bounds

Abstract

A vertex $v$ in a porous exponential dominating set assigns weight $\left(\tfrac{1}{2}\right)^{dist(v,u)}$ to vertex $u$. A porous exponential dominating set of a graph $G$ is a subset of $V(G)$ such that every vertex in $V(G)$ has been assigned a sum weight of at least 1. In this paper the porous exponential dominating number, denoted by $\gamma_e^*(G)$, for the graph $G = C_m \times C_n$ is discussed. Anderson et. al. proved that $\frac{mn}{15.875}\le \gamma_e^*(C_m \times C_n) \le \frac{mn}{13}$ and conjectured that $\frac{mn}{13}$ is also the asymptotic lower bound. We use a linear programing approach to sharpen the lower bound to $\frac{mn}{13.7619 + \epsilon(m,n)}$.