On the \(r\)-domination number of a graph
If \(r>0\) the \(r\)-domination number of a graph \(G_ n\) is the size \(d_ r\) of a smallest set of vertices such that every vertex of \(G\) is within distance \(r\) of a vertex in that set. The authors show, among other things, that if \(G\) has a spanning tree with at least \(n/2\) leaves then \(d_ r\leq\max\{n/(2r),1\}\). They conjecture that if the graph \(G_ n\) is Eulerian then \(d_ r\leq\lceil n/(2r)\rceil\).
scientific article; zbMATH DE number 89891
scientific article