Upper Broadcast Domination Number of Caterpillars with no Trunks

A broadcast on a graph G = ( V,E ) is a function f : V → { 0 , … , diam ( G ) } such that f ( v ) ≤ e G ( v ) for every vertex v ∈ V , where diam ( G ) denotes the diameter of G and e G ( v ) the eccentricity of v in G . Such a broadcast f is minimal if there does not exist any broadcast g ≠ f on G such that g ( v ) ≤ f ( v ) for all v ∈ V . The upper broadcast domination number of G is the maximum value of ∑ v ∈ V f ( v ) among all minimal broadcasts f on G for which each vertex of G is at distance at most f ( v ) from some vertex v with f ( v ) ≥ 1 . In this paper, we study the minimal dominating broadcasts of caterpillars and give the exact value of the upper broadcast domination number of caterpillars with no trunks.