Distance defined by spanning trees in graphs

• Autorzy: Gary Chartrand , Ladislav Nebeský , Ping Zhang
• Języki publikacji: EN

Abstrakty

EN

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 $d_{G|T}(u,v)$ 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 $d_{G|T}$ in graphs and present several realization results on parameters and subgraphs defined by these two distances.

Słowa kluczowe

EN

distance; geodesic; T-path; T-geodesic; T-distance

Kategorie tematyczne

• 05C12: Distance in graphs
• 05C05: Trees

• Wydawca: De Gruyter Open
• Czasopismo: Discussiones Mathematicae Graph Theory
• Rocznik: 2007
• Tom: 27
• Numer: 3
• Strony: 485-506

Daty

wydano

2007

otrzymano

2006-07-06

poprawiono

2007-04-18

zaakceptowano

2007-04-30

Twórcy

autor

Gary Chartrand

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

autor

Ladislav Nebeský

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

autor

Ping Zhang

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

Bibliografia

• [1] H. Bielak and M.M. Sysło, Peripheral vertices in graphs, Studia Sci. Math. Hungar. 18 (1983) 269-75.
• [2] F. Buckley, Z. Miller and P.J. Slater, On graphs containing a given graph as center, J. Graph Theory 5 (1981) 427-434, doi: 10.1002/jgt.3190050413.
• [3] G. Chartrand and P. Zhang, Introduction to Graph Theory (McGraw-Hill, Boston, 2005).
• [4] F. Harary and R.Z. Norman, The dissimilarity characteristic of Husimi trees, Ann. of Math. 58 (1953) 134-141, doi: 10.2307/1969824.
• [5] L. Nebeský, A characterization of the set of all shortest paths in a connected graph, Math. Bohem. 119 (1994) 15-20.
• [6] L. Nebeský, A new proof of a characterization of the set of all geodesics in a connected graph, Czech. Math. J. 48 (1998) 809-813, doi: 10.1023/A:1022404126392.
• [7] L. Nebeský, The set of geodesics in a graph, Discrete Math. 235 (2001) 323-326, doi: 10.1016/S0012-365X(00)00285-5.

Identyfikatory

DOI

10.7151/dmgt.1375