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A broadcast on a graph $G=(V,E)$ is a function $f:V \rightarrow \{0,1,\ldots, \text{diam}(G)\}$ satisfying $f(v) \leq e(v)$ for all $v \in V$, where$e(v)$ denotes the eccentricity of $v$ and $\text{diam}(G)$ denotes thediameter of $G$. We say that a broadcast dominates $G$ if every vertex can hearat least one broadcasting node. The upper domination number is the maximum costof all possible minimal broadcasts, where the cost of a broadcast is defined as$\text{cost} (f)= \sum_{v \in V}f(v)$. In this paper we establish both theupper domination number and the upper broadcast domination number on toroidalgrids. In addition, we classify all diametrical trees, that is, trees whoseupper domination number is equal to its diameter.

Upper broadcast domination of toroidal grids and a classification of diametrical trees