On the Broadcast Independence Number of Grid Graph

Abstract

A broadcast on a nontrivial connected graph G is a function ${f:V \longrightarrow \{0, \ldots,\operatorname{diam}(G)\}}$ such that for every vertex ${v \in V(G)}$ , ${f(v) \leq e(v)}$ , where ${\operatorname{diam}(G)}$ denotes the diameter of G and e(v) denotes the eccentricity of vertex v. The broadcast independence number is the maximum value of ${\sum_{v \in V} f(v)}$ over all broadcasts f that satisfy ${d(u,v) > \max \{f(u), f(v)\}}$ for every pair of distinct vertices u, v with positive values. We determine this invariant for grid graphs ${G_{m,n} = P_m \square P_n}$ , where ${2 \leq m \leq n}$ and □ denotes the Cartesian product. We hereby answer one of the open problems raised by Dunbar et al. in (Discrete Appl Math 154:59–75, 2006).