On exact \(n\)-step domination (Q1301849)

The author defines a set \(S\subset V\) as exact \(n\)-step dominating the graph \(G=(V,E)\) if for each vertex \(v\in V\) there is a unique vertex \(s\in S\) that has distance \(n\) to \(v\). Thus if \(S\) is an exact \(n\)-step dominating set, the graph is partitioned in metric spheres of radius \(n\) with centers in \(S\). This generalizes the exact 2-step domination of \textit{G. Chatrand}, \textit{F. Harary}, \textit{M. Hossain}, and \textit{K. Schultz} [Math. Bohem. 120, No.~2, 125-134 (1995; Zbl 0863.05050)]. The author proves some properties of exact \(n\)-step dominating sets and the underlying graphs, among them that all \(n\)-step dominating sets of a graph have the same cardinality and that the cardinality of an exact \(n\)-step dominating set is at least \(\lfloor \log_2 n\rfloor + 2\). Also the author obtains the following result which was conjectured in \textit{Y. Alavi, D. R. Lick} and \textit{H. B. Zou} [Second order degree regular graphs. Graph theory, combinatorics, and applications, Vol.~1, 1-8 (1991; Zbl 0841.05080)]: If each vertex in a connected graph \(G\) has exactly one vertex at distance \(n\) to it, then the diameter of \(G\) is \(n\), or \(G\) is a path \(P_{2n}\).