Distance defined by spanning trees in graphs

For a spanning tree T in a nontrivial connected graph G and for vertices u and v in G, there exists a unique u−v path u = u0, u1, u2, . . ., uk = v in T . A u− v T -path in G is a u− v path u = v0, v1, . . . , vl = v in G that is a subsequence of the sequence u = u0, u1, u2, . . . , uk = v. A u− v T -path of minimum length is a u− v T -geodesic in G. The T distance dG|T (u, v) from u to v in G is the length of a u−v T -geodesic. Let geo(G) and geo(G|T ) be the set of geodesics and the set of T geodesics respectively in G. Necessary and sufficient conditions are established for (1) geo(G) = geo(G|T ) and (2) geo(G|T ) = geo(G|T ), where T and T ∗ are two spanning trees of G. It is shown for a connected graph G that geo(G|T ) = geo(G) for every spanning tree T of G if and only if G is a block graph. For a spanning tree T of a connected graph G, it is also shown that geo(G|T ) satisfies seven of the eight axioms of the characterization of geo(G). Furthermore, we study the 486 G. Chartrand, L. Nebeský and P. Zhang relationship between the distance d and T -distance dG|T in graphs and present several realization results on parameters and subgraphs defined by these two distances.