# Leaps: an approach to the block structure of a graph

• Volume: 26, Issue: 1, page 77-90
• ISSN: 2083-5892

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## Abstract

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To study the block structure of a connected graph G = (V,E), we introduce two algebraic approaches that reflect this structure: a binary operation + called a leap operation and a ternary relation L called a leap system, both on a finite, nonempty set V. These algebraic structures are easily studied by considering their underlying graphs, which turn out to be block graphs. Conversely, we define the operation ${+}_{G}$ as well as the set of leaps ${L}_{G}$ of the connected graph G. The underlying graph of ${+}_{G}$, as well as that of ${L}_{G}$, turns out to be just the block closure of G (i.e., the graph obtained by making each block of G into a complete subgraph).

## How to cite

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Henry Martyn Mulder, and Ladislav Nebeský. "Leaps: an approach to the block structure of a graph." Discussiones Mathematicae Graph Theory 26.1 (2006): 77-90. <http://eudml.org/doc/270237>.

@article{HenryMartynMulder2006,
abstract = {To study the block structure of a connected graph G = (V,E), we introduce two algebraic approaches that reflect this structure: a binary operation + called a leap operation and a ternary relation L called a leap system, both on a finite, nonempty set V. These algebraic structures are easily studied by considering their underlying graphs, which turn out to be block graphs. Conversely, we define the operation $+_G$ as well as the set of leaps $L_G$ of the connected graph G. The underlying graph of $+_G$, as well as that of $L_G$, turns out to be just the block closure of G (i.e., the graph obtained by making each block of G into a complete subgraph).},
author = {Henry Martyn Mulder, Ladislav Nebeský},
journal = {Discussiones Mathematicae Graph Theory},
keywords = {leap; leap operation; block; cut-vertex; block closure; block graph},
language = {eng},
number = {1},
pages = {77-90},
title = {Leaps: an approach to the block structure of a graph},
url = {http://eudml.org/doc/270237},
volume = {26},
year = {2006},
}

TY - JOUR
AU - Henry Martyn Mulder
TI - Leaps: an approach to the block structure of a graph
JO - Discussiones Mathematicae Graph Theory
PY - 2006
VL - 26
IS - 1
SP - 77
EP - 90
AB - To study the block structure of a connected graph G = (V,E), we introduce two algebraic approaches that reflect this structure: a binary operation + called a leap operation and a ternary relation L called a leap system, both on a finite, nonempty set V. These algebraic structures are easily studied by considering their underlying graphs, which turn out to be block graphs. Conversely, we define the operation $+_G$ as well as the set of leaps $L_G$ of the connected graph G. The underlying graph of $+_G$, as well as that of $L_G$, turns out to be just the block closure of G (i.e., the graph obtained by making each block of G into a complete subgraph).
LA - eng
KW - leap; leap operation; block; cut-vertex; block closure; block graph
UR - http://eudml.org/doc/270237
ER -

## References

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1. [1] M. Changat, S. Klavžar and H.M. Mulder, The all-path transit function of a graph, Czechoslovak Math. J. 51 (2001) 439-448, doi: 10.1023/A:1013715518448. Zbl0977.05135
2. [2] F. Harary, A characterization of block graphs, Canad. Math. Bull. 6 (1963) 1-6, doi: 10.4153/CMB-1963-001-x. Zbl0112.25002
4. [4] M.A. Morgana and H.M. Mulder, The induced path convexity, betweenness, and svelte graphs, Discrete Math. 254 (2002) 349-370, doi: 10.1016/S0012-365X(01)00296-5. Zbl1003.05090
5. [5] H.M. Mulder, The interval function of a graph (MC-tract 132, Mathematish Centrum, Amsterdam 1980).
6. [6] H.M. Mulder, Transit functions on graphs, in preparation. Zbl1166.05019
7. [7] L. Nebeský, A characterization of the interval function of a graph, Czechoslovak Math. J. 44 (119) (1994) 173-178. Zbl0808.05046
8. [8] L. Nebeský, Geodesics and steps in a connected graph, Czechoslovak Math. J. 47 (122) (1997) 149-161. Zbl0898.05041
9. [9] L. Nebeský, An algebraic characterization of geodetic graphs, Czechoslovak Math. J. 48 (123) (1998) 701-710. Zbl0949.05022
10. [10] L. Nebeský, A tree as a finite nonempty set with a binary operation, Mathematica Bohemica 125 (2000) 455-458. Zbl0963.05032
11. [11] L. Nebeský, New proof of a characterization of geodetic graphs, Czechoslovak Math. J. 52 (127) (2002) 33-39. Zbl0995.05124

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