On the distance preserving trees in graphs

For a vertex v of a graph G, a spanning tree T of G is distancepreserving 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 Tu from u and Tv 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.