# A characterization of the set of all shortest paths in a connected graph

Mathematica Bohemica (1994)

- Volume: 119, Issue: 1, page 15-20
- ISSN: 0862-7959

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topNebeský, Ladislav. "A characterization of the set of all shortest paths in a connected graph." Mathematica Bohemica 119.1 (1994): 15-20. <http://eudml.org/doc/29362>.

@article{Nebeský1994,

abstract = {Let $G$ be a (finite undirected) connected graph (with no loop or multiple edge). The set $\mathcal \{L\}$ 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 $\mathcal \{L\}$ is based on determining the length of a path. Theorem 1 gives - metaphorically speaking - an “almost non-metric” characterization of $\mathcal \{L\}$: 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.},

author = {Nebeský, Ladislav},

journal = {Mathematica Bohemica},

keywords = {geodetic graphs; connected graph; shortest paths; geodetic graphs; connected graph; shortest paths},

language = {eng},

number = {1},

pages = {15-20},

publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},

title = {A characterization of the set of all shortest paths in a connected graph},

url = {http://eudml.org/doc/29362},

volume = {119},

year = {1994},

}

TY - JOUR

AU - Nebeský, Ladislav

TI - A characterization of the set of all shortest paths in a connected graph

JO - Mathematica Bohemica

PY - 1994

PB - Institute of Mathematics, Academy of Sciences of the Czech Republic

VL - 119

IS - 1

SP - 15

EP - 20

AB - Let $G$ be a (finite undirected) connected graph (with no loop or multiple edge). The set $\mathcal {L}$ 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 $\mathcal {L}$ is based on determining the length of a path. Theorem 1 gives - metaphorically speaking - an “almost non-metric” characterization of $\mathcal {L}$: 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.

LA - eng

KW - geodetic graphs; connected graph; shortest paths; geodetic graphs; connected graph; shortest paths

UR - http://eudml.org/doc/29362

ER -

## References

top- M. Behzad G. Chartrand, L. Lesniak-Foster, Graphs & Digraphs, Prindle, Weber & Schmidt, Boston, 1979. (1979) MR0525578
- D.C. Kay, G. Chartrand, 10.4153/CJM-1965-034-0, Canad. J. Math. 17 (1965), 342-346. (1965) Zbl0139.17301MR0175113DOI10.4153/CJM-1965-034-0
- H.M. Mulder, The Interval Function of a Graph, Mathematisch Centrum, Amsterdam, 1980. (1980) Zbl0446.05039MR0605838
- L. Nebeský, Route systems and bipartite graphs, Czechoslovak Math. Journal 41 (116) (1991), 260-264. (1991) MR1105440

## Citations in EuDML Documents

top- Ladislav Nebeský, A characterization of geodetic graphs
- Ladislav Nebeský, A characterization of the interval function of a connected graph
- Gary Chartrand, Ladislav Nebeský, Ping Zhang, Distance defined by spanning trees in graphs
- Ladislav Nebeský, On the set of all shortest paths of a given length in a connected graph
- Ladislav Nebeský, Visibilities and sets of shortest paths in a connected graph
- Ladislav Nebeský, An algebraic characterization of geodetic graphs
- Ladislav Nebeský, A new proof of a characterization of the set of all geodesics in a connected graph
- Ladislav Nebeský, Route systems of a connected graph
- Ladislav Nebeský, Geodesics and steps in a connected graph
- Ladislav Nebeský, On the distance function of a connected graph

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