Uniquely Radial Trees

A broadcast on a graph \( G \) is a function \( f: V \to \{0, 1, \dots, \text{diam}G\} \) such that \( f(v) < e(v) \) (the eccentricity of \( v \)) for all \( v \in V \). The broadcast number of \( G \) is the minimum value of \( \sum_{v \in V} f(v) \) among all broadcasts \( f \) for which each vertex of \( G \) is within distance \( f(v) \) from some vertex \( v \) with \( f(v) \geq 1 \). This number is bounded above by the radius of \( G \). A graph is uniquely radial if its only minimum broadcasts are broadcasts \( f \) such that \( f(v) = \text{rad}G \) for some central vertex \( v \), and \( f(u) = 0 \) if \( u \neq v \). We characterize uniquely radial trees.