Uniquely Radial Trees

Abstract

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 $f(v)\ge 1$. 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 $u\ne v$. We characterize uniquely radial trees.