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

TL;DR

This paper determines exact and optimal values for the 2-limited broadcast domination number on various grid graphs, using exhaustive case analysis combined with linear programming and mathematical proofs.

Contribution

It provides the first exact values for these parameters on multiple grid graph classes, advancing understanding of broadcast domination in grid structures.

Findings

Exact values for $C_m\square C_n$ with $3\leq m\leq 6$

Exact values for $P_m \square C_3$ for all $m\geq 3$

Periodically optimal values for certain grid graphs

Abstract

We establish exact values for the $2$-limited broadcast domination number of various grid graphs, in particular $C_m\square C_n$ for $3 \leq m \leq 6$ and all $n\geq m$, $P_m \square C_3$ 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 optimal values for $P_m \square C_4$ and $P_m \square C_6$ for $m \geq 3$, $P_4 \square P_n$ for $n \geq 4$, and $P_5 \square P_n$ for $n \geq 5$. Our method completes an exhaustive case analysis and eliminates cases by combining tools from linear programming with various mathematical proof techniques.