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

TL;DR

This paper investigates the maximum minimal broadcast cost in toroidal grids and classifies trees where the upper domination number equals the diameter, advancing understanding of broadcast domination in graphs.

Contribution

It determines the upper broadcast domination number for toroidal grids and classifies all diametrical trees, providing new insights into graph domination parameters.

Findings

Established the upper broadcast domination number for toroidal grids.

Classified all diametrical trees where upper domination equals diameter.

Provided formulas and characterizations for these graph classes.

Abstract

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 the diameter of $G$. We say that a broadcast dominates $G$ if every vertex can hear at least one broadcasting node. The upper domination number is the maximum cost of 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 the upper domination number and the upper broadcast domination number on toroidal grids. In addition, we classify all diametrical trees, that is, trees whose upper domination number is equal to its diameter.