| Title:
|
A characterization of the set of all shortest paths in a connected graph (English) |
| Author:
|
Nebeský, Ladislav |
| Language:
|
English |
| Journal:
|
Mathematica Bohemica |
| ISSN:
|
0862-7959 (print) |
| ISSN:
|
2464-7136 (online) |
| Volume:
|
119 |
| Issue:
|
1 |
| Year:
|
1994 |
| Pages:
|
15-20 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Let $G$ be a (finite undirected) connected graph (with no loop or multiple edge). The set $\Cal L$ of all shortest paths in $G$ is defined as the set of all paths $\xi$, then the lenght of $\xi$ does not exceed the length of $\varsigma$. While the definition of $\Cal L$ is based on determining the length of a path. Theorem 1 gives - metaphorically speaking - an "almost non-metric" characterization of $\Cal L$: a characterization in which the length of a path greater than one is not considered. Two other theorems are derived from Theorem 1. One of them (Theorem 3) gives a characterization of geodetic graphs. (English) |
| Keyword:
|
geodetic graphs |
| Keyword:
|
connected graph |
| Keyword:
|
shortest paths |
| MSC:
|
05C12 |
| MSC:
|
05C38 |
| MSC:
|
05C75 |
| idZBL:
|
Zbl 0807.05045 |
| idMR:
|
MR1303548 |
| DOI:
|
10.21136/MB.1994.126208 |
| . |
| Date available:
|
2009-09-24T21:02:18Z |
| Last updated:
|
2020-07-29 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/126208 |
| . |
| Reference:
|
[1] M. Behzad G. Chartrand, L. Lesniak-Foster: Graphs & Digraphs.Prindle, Weber & Schmidt, Boston, 1979. MR 0525578 |
| Reference:
|
[2] D.C. Kay, G. Chartrand: A characterization of certain ptolemaic graphs.Canad. J. Math. 17 (1965), 342-346. Zbl 0139.17301, MR 0175113, 10.4153/CJM-1965-034-0 |
| Reference:
|
[3] H.M. Mulder: The Interval Function of a Graph.Mathematisch Centrum, Amsterdam, 1980. Zbl 0446.05039, MR 0605838 |
| Reference:
|
[4] L. Nebeský: Route systems and bipartite graphs.Czechoslovak Math. Journal 41 (116) (1991), 260-264. MR 1105440 |
| . |
