The Connected Domination Number of Grids

TL;DR

This paper investigates the connected domination number of grid graphs, providing new lower bounds and analyzing the optimality of existing constructions, advancing understanding of grid domination problems.

Contribution

It introduces a new lower bound for the connected domination number of grid graphs and examines the optimality of a known construction.

Findings

New lower bound of rac{mn+2rac{\u00c7min rac{m,nrac{3}{}}}{3} for arbitrary m,n  4.

Analysis of Fujie's construction for connected dominating sets.

Comparison showing the new bound improves previous bounds for large grids.

Abstract

Closed form expressions for the domination number of an $n \times m$ grid have attracted significant attention, and an exact expression has been obtained in 2011 by Gon\c{c}alves et al. In this paper, we present our results on obtaining new lower bounds on the connected domination number of an $n \times m$ grid. The problem has been solved for grids with up to $4$ rows and with $6$ rows by Tolouse et al and the best currently known lower bound for arbitrary $m,n$ is $\lceil\frac{mn}{3}\rceil$. Fujie came up with a general construction for a connected dominating set of an $n \times m$ grid of size $\min \left\{2n+(m-4)+\lfloor\frac{m-4}{3}\rfloor(n-2), 2m+(n-4)+\lfloor\frac{n-4}{3}\rfloor(m-2) \right\}$ . In this paper, we investigate whether this construction is indeed optimum. We prove a new lower bound of $\left\lceil\frac{mn+2\lceil\frac{\min \{m,n\}}{3}\rceil}{3} \right\rceil$ for arbitrary $m,n \geq 4$.