Broadcasts in Graphs: Diametrical Trees

TL;DR

This paper investigates trees where the upper broadcast domination number equals the diameter, providing characterizations of such trees including caterpillars, and explores bounds on broadcast domination in graphs.

Contribution

It characterizes trees, especially caterpillars, where the upper broadcast domination number equals the diameter, advancing understanding of broadcast domination in graph theory.

Findings

Identifies trees with upper broadcast domination number equal to their diameter.

Provides characterizations of caterpillars with this property.

Establishes bounds on broadcast domination numbers in graphs.

Abstract

A dominating broadcast on a graph G with vertex set V is a function f that maps V to {0,1,...,diam(G)} such that f(v) does not exceed e(v) (the eccentricity of v) for all vertices v, and each vertex u is at distance at most f(v) from a vertex v with positive f(v). The upper broadcast domination number of G is {\Gamma}_{b}(G), which equals the maximum of the sum of the function values f(v), the maximum being taken over all minimal dominating broadcasts f on G. As shown by Erwin in [D. Erwin, Cost domination in graphs, Doctoral dissertation, Western Michigan University, 2001], {\Gamma}_{b}(G) is bounded below by diam(G) for any graph G. We investigate trees whose upper broadcast domination number equal their diameter and, among more general results, characterize caterpillars with this property.