# An algebraic characterization of geodetic graphs

• Volume: 48, Issue: 4, page 701-710
• ISSN: 0011-4642

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## Abstract

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We say that a binary operation $*$ is associated with a (finite undirected) graph $G$ (without loops and multiple edges) if $*$ is defined on $V\left(G\right)$ and $uv\in E\left(G\right)$ if and only if $u\ne v$, $u*v=v$ and $v*u=u$ for any $u$, $v\in V\left(G\right)$. In the paper it is proved that a connected graph $G$ is geodetic if and only if there exists a binary operation associated with $G$ which fulfils a certain set of four axioms. (This characterization is obtained as an immediate consequence of a stronger result proved in the paper).

## How to cite

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Nebeský, Ladislav. "An algebraic characterization of geodetic graphs." Czechoslovak Mathematical Journal 48.4 (1998): 701-710. <http://eudml.org/doc/30448>.

@article{Nebeský1998,
abstract = {We say that a binary operation $*$ is associated with a (finite undirected) graph $G$ (without loops and multiple edges) if $*$ is defined on $V(G)$ and $uv\in E(G)$ if and only if $u\ne v$, $u * v=v$ and $v*u=u$ for any $u$, $v\in V(G)$. In the paper it is proved that a connected graph $G$ is geodetic if and only if there exists a binary operation associated with $G$ which fulfils a certain set of four axioms. (This characterization is obtained as an immediate consequence of a stronger result proved in the paper).},
journal = {Czechoslovak Mathematical Journal},
keywords = {geodetic graph; binary operation; connected graph},
language = {eng},
number = {4},
pages = {701-710},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {An algebraic characterization of geodetic graphs},
url = {http://eudml.org/doc/30448},
volume = {48},
year = {1998},
}

TY - JOUR
TI - An algebraic characterization of geodetic graphs
JO - Czechoslovak Mathematical Journal
PY - 1998
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 48
IS - 4
SP - 701
EP - 710
AB - We say that a binary operation $*$ is associated with a (finite undirected) graph $G$ (without loops and multiple edges) if $*$ is defined on $V(G)$ and $uv\in E(G)$ if and only if $u\ne v$, $u * v=v$ and $v*u=u$ for any $u$, $v\in V(G)$. In the paper it is proved that a connected graph $G$ is geodetic if and only if there exists a binary operation associated with $G$ which fulfils a certain set of four axioms. (This characterization is obtained as an immediate consequence of a stronger result proved in the paper).
LA - eng
KW - geodetic graph; binary operation; connected graph
UR - http://eudml.org/doc/30448
ER -

## References

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1. Graphs & Digraphs, Prindle, Weber & Schmidt, Boston, 1979. (1979) MR0525578
3. The Interval Function of a Graph, Mathematisch Centrum. Amsterdam, 1980. (1980) Zbl0446.05039MR0605838
4. A characterization of the set of all shortest paths in a connected graph, Mathematica Bohemica 119 (1994), 15–20. (1994) MR1303548
5. A characterization of geodetic graphs, Czechoslovak Math. Journal 45 (120) (1995), 491–493. (1995) MR1344515
6. 10.1023/A:1022404624515, Czechoslovak Math. Journal 47 (122) (1997), 149–161. (1997) MR1435613DOI10.1023/A:1022404624515

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