ISSN 1234-3099 (print version)

ISSN 2083-5892 (electronic version)

Discussiones Mathematicae Graph Theory

IMPACT FACTOR 2018: 0.741

SCImago Journal Rank (SJR) 2018: 0.763

Rejection Rate (2017-2018): c. 84%

Discussiones Mathematicae Graph Theory


Discussiones Mathematicae Graph Theory 26(1) (2006) 135-140
DOI: 10.7151/dmgt.1307


Wilfried Imrich

Department of Mathematics and Information Technology
Montanuniversität Leoben
Franz Josef-Straße 18, A-8700 Leoben, Austria

Peter F. Stadler

Bioinformatics Group, Department of Computer Science
University of Leipzig
Härtelstrasse 16-18, D-04107 Leipzig, Germany
The Santa Fe Institute
1399 Hyde Park Road, Santa Fe, NM 87501, USA


We introduce the concept of neighborhood systems as a generalization of directed, reflexive graphs and show that the prime factorization of neighborhood systems with respect to the the direct product is unique under the condition that they satisfy an appropriate notion of thinness.

Keywords: products, set systems, prime factor theorem.

2000 Mathematics Subject Classification: 05C20, 05C65, 05C70.


[1] E. Cech, Topological spaces, Revised edition by Z. Frolí k and M. Katetov, Scientific editor, V. Pták. Editor of the English translation, Charles O. Junge (Publishing House of the Czechoslovak Academy of Sciences, Prague, 1966).
[2] W. Fontana and P. Schuster, Continuity in Evolution: On the Nature of Transitions, Science 280 (1998) 1451-1455, doi: 10.1126/science.280.5368.1451.
[3] P.C. Hammer, Extended topology: Continuity. I, Portugal. Math. 23 (1964) 77-93.
[4] W. Imrich and S. Klavžar, Product Graphs, Wiley-Interscience Series in Discrete Mathematics and Optimization, (Wiley-Interscience, New York, 2000) Structure and recognition, With a foreword by P. Winkler.
[5] R. McKenzie, Cardinal multiplication of structures with a reflexive relation, Fund. Math. 70 (1971) 59-101.
[6] B.M.R. Stadler and P.F. Stadler, Generalized topological spaces in evolutionary theory and combinatorial chemistry, J. Chem. Inf. Comput. Sci. 42 (2002) 577-585, doi: 10.1021/ci0100898.
[7] B.M.R. Stadler, P.F. Stadler, G. Wagner and W. Fontana, The topology of the possible: Formal spaces underlying patterns of evolutionary change, J. Theor. Biol. 213 (2001) 241-274, doi: 10.1006/jtbi.2001.2423.
[8] G. Wagner and P.F. Stadler, Quasi-independence, homology and the unity of type: A topological theory of characters, J. Theor. Biol. 220 (2003) 505-527, doi: 10.1006/jtbi.2003.3150.

Received 31 March 2005
Revised 7 September 2005