Redundance of Complete Grid Graphs

The redundance \( R(G) \) of a graph \( G \) is the minimum, over all dominating sets \( S \), of \( \sum_{v \in S} (1 + \deg(v)) \), where \( \deg(v) \) is the degree of vertex \( v \). We use some dynamic programming algorithms to compute the redundance of complete grid graphs \( G_{m,n} \) for \( 1 \leq m \leq 21 \) and all \( n \), and to establish good upper and lower bounds on the redundance for larger \( m \). We conjecture that the upper bound is the redundance when \( m > 21 \).