Domination Numbers of Complete Grid Graphs, $I$

Abstract

This paper concerns the domination numbers ${\gamma }_{k,n}$ for the complete $k\times n$ grid graphs for $1\le k\le 10$ and $n\ge 1$. These numbers were previously established for $1\le k\le 4$. Here we present dominating sets for $5\le k\le 10$ and $n\ge 1$. This gives upper bounds for ${\gamma }_{k,n}$ for $k$ in this range. We discuss evidence that indicates that these upper bounds are also lower bounds.