On the distance preserving trees in graphs

For a vertex $v$ of a graph $G$, a spanning tree $T$ of $G$ is distance-preserving from $v$ if, for any vertex $w$, the distance from $v$ to $w$ on $T$ is the same as the distance from $v$ to $w$ on $G$. If two vertices $u$ and $v$ are distinct, then two distance-preserving spanning trees $T_u$ from $u$ and $T_v$ from $v$ are distinct in general. A purpose of this paper is to give a characterization for a given weighted graph $G$ to have a spanning tree $T$ such that $T$ is a distance-preserving spanning tree from distinct two vertices.