Radial trees

A broadcast on a graph G is a function such that for each v∈V, f(v)≤e(v) (the eccentricity of v). The broadcast number of G is the minimum value of ∑_v∈Vf(v) among all broadcasts f for which each vertex of G is within distance f(v) from some vertex v having f(v)≥1. This number is bounded above by the radius of G as well as by its domination number. Graphs for which the broadcast number is equal to the radius are called radial; the problem of characterizing radial trees was first discussed in J. Dunbar, D. Erwin, T. Haynes, S.M. Hedetniemi, S.T. Hedetniemi, Broadcasts in graphs, Discrete Appl. Math. (154) (2006) 59-75].We provide a characterization of radial trees as well as a geometrical interpretation of our characterization.