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Duality of Ends

Published online by Cambridge University Press:  10 August 2009

HENNING BRUHN  and
MAYA STEIN
HENNING BRUHN
Affiliation:
Mathematisches Seminar, Universität Hamburg, Bundesstraße 55, 20146 Hamburg, Germany (e-mail: [email protected])
MAYA STEIN
Affiliation:
Centro de Modelamiento Matemático, Universidad de Chile, Blanco Encalada 2120, Santiago, Chile (e-mail: [email protected])
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Abstract

We investigate the end spaces of infinite dual graphs. We show that there exists a natural homeomorphism * between the end spaces of a graph and its dual, and that * preserves the ‘end degree’. In particular, * maps thick ends to thick ends. Along the way, we prove that Tutte-connectivity is invariant under taking (infinite) duals.

Type
Paper
Information
Combinatorics, Probability and Computing , Volume 19 , Issue 1 , January 2010 , pp. 47 - 60
Copyright
Copyright © Cambridge University Press 2009

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References

[1]Bruhn, H. and Diestel, R. (2006) Duality in infinite graphs. Combin. Probab. Comput. 15 75–90.CrossRefGoogle Scholar
[2]Bruhn, H. and Stein, M. (2007) On end degrees and infinite circuits in locally finite graphs. Combinatorica 27 269–291.CrossRefGoogle Scholar
[3]Diestel, R. (2005) The cycle space of an infinite graph. Combin. Probab. Comput. 14 59–79.CrossRefGoogle Scholar
[4]Diestel, R. (2005) Graph Theory, 3rd edn, Springer.Google Scholar
[5]Diestel, R. and Kühn, D. (2004) On infinite cycles I. Combinatorica 24 69–89.CrossRefGoogle Scholar
[6]Diestel, R. and Kühn, D. (2004) On infinite cycles II. Combinatorica 24 91–116.CrossRefGoogle Scholar
[7]Diestel, R. and Kühn, D. (2004) Topological paths, cycles and spanning trees in infinite graphs. Europ. J. Combin. 25 835–862.CrossRefGoogle Scholar
[8]Hahn, G., Laviolette, F. and Širáň, J. (1997) Edge-ends in countable graphs. J. Combin. Theory Ser. B 70 225–244.CrossRefGoogle Scholar
[9]Halin, R. (1965) Über die Maximalzahl fremder unendlicher Wege in Graphen. Math. Nachr. 30 63–85.CrossRefGoogle Scholar
[10]Higgs, D. A. (1969) Matroids and duality. Colloq. Math. 20 215–220.CrossRefGoogle Scholar
[11]Oxley, J. G. (1992) Infinite matroids. In Matroid Applications (White, N., ed.), Vol. 40 of Encyclopedia of Mathematics and its Applications, Cambridge University Press, pp. 73–90.CrossRefGoogle Scholar
[12]Polat, N. (1991) A Mengerian theorem for infinite graphs with ideal points. J. Combin. Theory Ser. B 51 248–255.CrossRefGoogle Scholar
[13]Stein, M. (2007) Forcing highly connected subgraphs. J. Graph Theory 54 331–349.CrossRefGoogle Scholar
[14]Thomassen, C. (1980) Planarity and duality of finite and infinite graphs. J. Combin. Theory Ser. B 29 244–271.CrossRefGoogle Scholar
[15]Thomassen, C. (1982) Duality of infinite graphs. J. Combin. Theory Ser. B 33 137–160.CrossRefGoogle Scholar
[16]Thomassen, C. and Woess, W. (1993) Vertex-transitive graphs and accessibility. J. Combin. Theory Ser. B 58 248–268.CrossRefGoogle Scholar
[17]Tutte, W. T. (1966) Connectivity in matroids. Canad. J. Math. 18 1301–1324.CrossRefGoogle Scholar
[18]Vella, A. and Richter, R. B. (2008) Cycle spaces of topological spaces. J. Graph Theory 59 115–144.CrossRefGoogle Scholar
[19]Whitney, H. (1932) Non-separable and planar graphs. Trans. Amer. Math. Soc. 34 339–362.CrossRefGoogle Scholar