Solutions of a problem of Ore on spanning trees and its generalization

We consider graphs, not necessarily finite, with neither loops nor multiple edges. Pertinent definitions are given below. For notation and definitions not given here, we generally follow Harary and Buckley [1]. Let G = (V,E) be a connected graph. The distance between vertices u and v in G, dG(u,v), is the shortest length of a u - v path in G. The eccentricity of u in G, eG(u) = sup{dG(u,v) : v ∈ V(G)}. A spanning tree T of a graph G is called a distance preserving spanning tree of G if there exists a vertex uT of G such that dT(uT,x) = dG(uT,x) for every vertex x of G. Note that for every vertex v of a connected graph G there exists a distance preserving spanning tree T of G with uT = v. Every subtree of G can be extended to a spanning tree of G, by the axiom of choice. A connected graph G is said to have the property P if every spanning tree T of G is a distance preserving spanning tree of G. The following problem was posed by Ore [see [2], page 102, problem 4].