Proving exact values for the $2$-limited broadcast domination number on grid graphs

We establish exact values for the $2$-limited broadcast domination number ofvarious grid graphs, in particular $C_m\square C_n$ for $3 \leq m \leq 6$ andall $n\geq m$, $P_m \square C_3$ for all $m \geq 3$, and $P_m \square C_n$ for$4\leq m \leq 5$ and all $n \geq m$. We also produce periodically optimalvalues for $P_m \square C_4$ and $P_m \square C_6$ for $m \geq 3$, $P_4 \squareP_n$ for $n \geq 4$, and $P_5 \square P_n$ for $n \geq 5$. Our method completesan exhaustive case analysis and eliminates cases by combining tools from linearprogramming with various mathematical proof techniques