Broadcasts in Graphs: Diametrical Trees

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.