Radial trees

A broadcast on a graph G is a function f : V → { 0 , … , diam G } 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 ∈ V f ( 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]. vide a characterization of radial trees as well as a geometrical interpretation of our characterization.