The Connected Domination Number of Grids

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 Gonçalves 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 \(m,n \geq 4\).