Discussiones Mathematicae Graph Theory 27(3) (2007) 485-506
doi: 10.7151/dmgt.1375

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DISTANCE DEFINED BY SPANNING TREES IN GRAPHS

Gary Chartrand

Department of Mathematics
Western Michigan University
Kalamazoo, MI 49008, USA

Ladislav Nebeský

Faculty of Philosophy  & Arts
Charles University, Prague
J. Palacha 2, CZ - 116 38 Praha 1, Czech Republic

Ping Zhang

Department of Mathematics
Western Michigan University
Kalamazoo, MI 49008, USA

Abstract

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 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.

Keywords: distance, geodesic, T-path, T-geodesic, T-distance.

2000 Mathematics Subject Classification: 05C05, 05C12.

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Received 6 July 2006
Revised 18 April 2007
Accepted 30 April 2007