DMGT

ISSN 1234-3099 (print version)

ISSN 2083-5892 (electronic version)

https://doi.org/10.7151/dmgt

Discussiones Mathematicae Graph Theory

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Discussiones Mathematicae Graph Theory

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Discussiones Mathematicae Graph Theory 26(1) (2006) 77-90
DOI: 10.7151/dmgt.1303

LEAPS: AN APPROACH TO THE BLOCK STRUCTURE OF A GRAPH

Henry Martyn Mulder

Econometrisch Instituut, Erasmus Universiteit
P.O. Box 1738, 3000 DR Rotterdam, The Netherlands
e-mail: hmmulder@few.eur.nl

Ladislav Nebeský

Filozofická Fakulta, Univerzita Karlova v Praze
J. Palacha 2, 116 38 Praha 1, Czech Republic
e-mail: Ladislav.Nebesky@ff.cuni.cz

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 LG of the connected graph G. The underlying graph of +G, as well as that of LG, 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).

Keywords: leap, leap operation, block, cut-vertex, block closure, block graph.

2000 Mathematics Subject Classification: 05C99, 05C75, 08A99.

References

[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.
[2] F. Harary, A characterization of block graphs, Canad. Math. Bull. 6 (1963) 1-6, doi: 10.4153/CMB-1963-001-x.
[3] F. Harary, Graph Theory (Addison-Wesley, Reading MA, 1969).
[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.
[5] H.M. Mulder, The interval function of a graph (MC-tract 132, Mathematish Centrum, Amsterdam 1980).
[6] H.M. Mulder, Transit functions on graphs, in preparation.
[7] L. Nebeský, A characterization of the interval function of a graph, Czechoslovak Math. J. 44 (119) (1994) 173-178.
[8] L. Nebeský, Geodesics and steps in a connected graph, Czechoslovak Math. J. 47 (122) (1997) 149-161.
[9] L. Nebeský, An algebraic characterization of geodetic graphs, Czechoslovak Math. J. 48 (123) (1998) 701-710.
[10] L. Nebeský, A tree as a finite nonempty set with a binary operation, Mathematica Bohemica 125 (2000) 455-458.
[11] L. Nebeský, New proof of a characterization of geodetic graphs, Czechoslovak Math. J. 52 (127) (2002) 33-39.

Received 14 January 2005
Revised 20 June 2005