Radial trees

A broadcast   on a graph GG is a function f:V→{0,…,diamG} such that for each v∈Vv∈V, f(v)≤e(v)f(v)≤e(v) (the eccentricity of vv). The broadcast number   of GG is the minimum value of ∑v∈Vf(v)∑v∈Vf(v) among all broadcasts ff for which each vertex of GG is within distance f(v)f(v) from some vertex vv having f(v)≥1f(v)≥1. This number is bounded above by the radius of GG 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.