| Title:
|
On properties of a graph that depend on its distance function (English) |
| Author:
|
Nebeský, Ladislav |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
54 |
| Issue:
|
2 |
| Year:
|
2004 |
| Pages:
|
445-456 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
If $G$ is a connected graph with distance function $d$, then by a step in $G$ is meant an ordered triple $(u, x, v)$ of vertices of $G$ such that $d(u, x) = 1$ and $d(u, v) = d(x, v) + 1$. A characterization of the set of all steps in a connected graph was published by the present author in 1997. In Section 1 of this paper, a new and shorter proof of that characterization is presented. A stronger result for a certain type of connected graphs is proved in Section 2. (English) |
| Keyword:
|
connected graphs |
| Keyword:
|
distance |
| Keyword:
|
steps |
| Keyword:
|
geodetically smooth graphs |
| MSC:
|
05C12 |
| MSC:
|
05C75 |
| idZBL:
|
Zbl 1080.05506 |
| idMR:
|
MR2059265 |
| . |
| Date available:
|
2009-09-24T11:14:21Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/127902 |
| . |
| Reference:
|
[1] H.-J. Bandelt and H. M. Mulder: Pseudo-modular graphs.Discrete Math. 62 (1986), 245–260. MR 0866940, 10.1016/0012-365X(86)90212-8 |
| Reference:
|
[2] G. Chartrand and L. Lesniak: Graphs & Digraphs. Third edition.Chapman & Hall, London, 1996. MR 1408678 |
| Reference:
|
[3] S. Klavžar and H. M. Mulder: Median graphs: characterizations, location theory and related structures.J. Combin. Math. Combin. Comput. 30 (1999), 103–127. MR 1705337 |
| Reference:
|
[4] H. M. Mulder: The interval function of a graph.Math. Centre Tracts 132, Math. Centre, Amsterdam, 1980. Zbl 0446.05039, MR 0605838 |
| Reference:
|
[5] H. M. Mulder and L. Nebeský: Modular and median signpost systems and their underlying graphs.Discuss. Math. Graph Theory 23 (2003), 309–32444. MR 2070159, 10.7151/dmgt.1204 |
| Reference:
|
[6] L. Nebeský: Geodesics and steps in a connected graph.Czechoslovak Math. J. 47 (122) (1997), 149–161. MR 1435613, 10.1023/A:1022404624515 |
| Reference:
|
[7] L. Nebeský: An axiomatic approach to metric properties of connected graphs.Czechoslovak Math. J. 50 (125) (2000), 3–14. MR 1745453, 10.1023/A:1022472700080 |
| Reference:
|
[8] L. Nebeský: A theorem for an axiomatic approach to metric properties of graphs.Czechoslovak Math. J. 50 (125) (2000), 121–133. MR 1745467, 10.1023/A:1022401506441 |
| Reference:
|
[9] L. Nebeský: A tree as a finite nonempty set with a binary operation.Math. Bohem. 125 (2000), 455–458. MR 1802293 |
| . |
