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 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 relationship between the distance d and T-distance in graphs and present several realization results on parameters and subgraphs defined by these two distances.
@article{bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1375,
author = {Gary Chartrand and Ladislav Nebesk\'y and Ping Zhang},
title = {Distance defined by spanning trees in graphs},
journal = {Discussiones Mathematicae Graph Theory},
volume = {27},
year = {2007},
pages = {485-506},
zbl = {1142.05013},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1375}
}
Gary Chartrand; Ladislav Nebeský; Ping Zhang. Distance defined by spanning trees in graphs. Discussiones Mathematicae Graph Theory, Tome 27 (2007) pp. 485-506. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1375/
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