Graphs having distance-\(n\) domination number half their order (Q1613368)

A set \(D\) of vertices in a graph \(G\) is distance-\(n\) dominating if, for every vertex \(v\in V(G)- D\), there is a vertex \(d\in D\) such that \(\text{dist}(v, d)= n\). The authors give a characterization of graphs \(G\) of diameter at least \(2n-1\) that have a minimum distance-\(n\) dominating set of cardinality \(p/2\), where \(p\) is the order of \(G\). This confirms a conjecture by James W. Boland, Teresa W. Haynes and Linda M. Lawson.