I am looking for solution of the following problem.
Let $G$ be a weighted graph with (positive) weights. The length of a path in a weighted graph is the sum of the weights of the selected edges. The distance between two nodes is a minimal length of path between these nodes. If $N$ is a node in $G$ and $R > O$, we can define a circle with center at $N$ and radius $R$, or $N,R$-circle, as a set of all nodes in $G$, whose distance to node $N$ is $\leq R$; we say that all these nodes are covered by this $N,R$-circle.
Question: what is a minimum number of circles of radius $R$ that cover all nodes of the graph $G$.
In my case $G$ is a tree with a finite number of nodes and edges, may be it simplifies the problem?