Small step-dominating sets in trees (Q870988)

The paper proves that for every tree \(T\) of diameter \(D\geq 3\) there is a set \(S\subseteq V(T)\) with \(| S| =D-1\) and a mapping st \(: S\rightarrow \{0,1,2,...\} \) such that for every vertex \(v\in V(T)\) there is exactly one vertex \(u\in S\) whose distance to \(v\) equals st\((u)\). This settles a conjecture of \textit{G. Dror, A. Lev}, and \textit{Y. Roditty} [Discrete Math. 289, 137--144 (2004; Zbl 1055.05112)].