A linear programming method for exponential domination

TL;DR

This paper introduces a linear programming approach to determine bounds on the porous exponential domination number in various graphs, including grids and hypercubes, advancing understanding of exponential domination metrics.

Contribution

It develops a linear programming method to compute bounds on the porous exponential domination number, improving previous bounds for specific graph classes.

Findings

Lower and upper bounds for $\gamma_e^*(G)$ in grid graphs and hypercubes.

A technique to derive lower bounds for the porous exponential domination number.

Application of linear programming to sharpen bounds for exponential domination.

Abstract

For a graph $G,$ the set $D \subseteq V(G)$ is a porous exponential dominating set if $1 \le \sum_{d \in D} \left( 2 \right)^{1-dist(d,v)}$ for every $v \in V(G),$ where $dist(d,v)$ denotes the length of the shortest $dv$ path. The porous exponential dominating number of $G,$ denoted $\gamma_e^*(G),$ is the minimum cardinality of a porous exponential dominating set. For any graph $G,$ a technique is derived to determine a lower bound for $\gamma_e^*(G).$ Specifically for a grid graph $H,$ linear programing is used to sharpen bound found through the lower bound technique. Lower and upper bounds are determined for the porous exponential domination number of the King Grid $\mathcal{K_n},$ the Slant Grid $\mathcal{S_n},$ and the $n$-dimensional hypercube $Q_n.$