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The Degree Sequence of Random Graphs from Subcritical Classes†

Published online by Cambridge University Press:  01 September 2009

NICLA BERNASCONI ,
KONSTANTINOS PANAGIOTOU  and
ANGELIKA STEGER
NICLA BERNASCONI
Affiliation:
Institute of Theoretical Computer Science, ETH Zürich, Universitätsstrasse 6, CH-8092 Zürich, Switzerland (e-mail: [email protected])
KONSTANTINOS PANAGIOTOU
Affiliation:
Institute of Theoretical Computer Science, ETH Zürich, Universitätsstrasse 6, CH-8092 Zürich, Switzerland (e-mail: [email protected])
ANGELIKA STEGER
Affiliation:
Institute of Theoretical Computer Science, ETH Zürich, Universitätsstrasse 6, CH-8092 Zürich, Switzerland (e-mail: [email protected])
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Abstract

In this work we determine the expected number of vertices of degree k = k(n) in a graph with n vertices that is drawn uniformly at random from a subcritical graph class. Examples of such classes are outerplanar, series-parallel, cactus and clique graphs. Moreover, we provide exponentially small bounds for the probability that the quantities in question deviate from their expected values.

Type
Paper
Information
Combinatorics, Probability and Computing , Volume 18 , Issue 5: New Directions in Algorithms, Combinatorics and Optimization , September 2009 , pp. 647 - 681
Copyright
Copyright © Cambridge University Press 2009

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References

[1]Bender, E. A., Gao, Z. and Wormald, N. C. (2002) The number of labeled 2-connected planar graphs. Electron. J. Combin. 9 #43.CrossRefGoogle Scholar
[2]Bergeron, F., Labelle, G. and Leroux, P. (1998) Combinatorial Species and Tree-Like Structures, Vol. 67 of Encyclopedia of Mathematics and its Applications, Cambridge University Press.Google Scholar
[3]Bernasconi, N., Panagiotou, K. and Steger, A. (2008) On properties of random dissections and triangulations. In Proc. 19th Annual ACM–SIAM Symposium on Discrete Algorithms (SODA ’08), pp. 132–141. Full version submitted Dec ’07, available at: www.as.inf.ethz.ch/research/publications/2008/index/.Google Scholar
[4]Bodirsky, M., Giménez, O., Kang, M. and Noy, M. (2005) On the number of series parallel and outerplanar graphs. In 2005 European Conference on Combinatorics, Graph Theory and Applications (EuroComb ’05), Vol. AE of DMTCS Proceedings, pp. 383–388.CrossRefGoogle Scholar
[5]Bollobás, B. (2001) Random Graphs, 2nd edn, Vol. 73 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge.CrossRefGoogle Scholar
[6]Chapuy, G., Fusy, É., Kang, M. and Shoilekova, B. (2008) A complete grammar for decomposing a family of graphs into 3-connected components. Electron. J. Combin. 15 #148.CrossRefGoogle Scholar
[7]Denise, A., Vasconcellos, M. and Welsh, D. J. A. (1996) The random planar graph. Congr. Numer. 113 61–79.Google Scholar
[8]Drmota, M., Giménez, O. and Noy, M. Vertices of given degree in series-parallel graphs. Random Structures and Algorithms. To appear.Google Scholar
[9]Duchon, P., Flajolet, P., Louchard, G. and Schaeffer, G. (2004) Boltzmann samplers for the random generation of combinatorial structures. Combin. Probab. Comput. 13 577–625.CrossRefGoogle Scholar
[10]Flajolet, P. and Sedgewick, R. (2009) Analytic Combinatorics, Cambridge University Press.CrossRefGoogle Scholar
[11]Flajolet, P., Zimmerman, P. and Van Cutsem, B. (1994) A calculus for the random generation of labelled combinatorial structures. Theoret. Comput. Sci. 132 1–35.CrossRefGoogle Scholar
[12]Fusy, É. (2005) Quadratic exact-size and linear approximate-size random generation of planar graphs. In 2005 International Conference on Analysis of Algorithms (Martínez, C., ed.), Vol. AD of DMTCS Proceedings, pp. 125–138.CrossRefGoogle Scholar
[13]Gerke, S., McDiarmid, C., Steger, A. and Weißl, A. (2005) Random planar graphs with n nodes and a fixed number of edges. In Proc. 16th Annual ACM–SIAM Symposium on Discrete Algorithms, pp. 999–1007, ACM, New York.Google Scholar
[14]Giménez, O. and Noy, M. (2005) The number of planar graphs and properties of random planar graphs. In 2005 International Conference on Analysis of Algorithms, Vol. AD of DMTCS Proceedings, pp. 147–156.CrossRefGoogle Scholar
[15]Janson, S., Łuczak, T. and Ruciński, A. (2000) Random Graphs, Wiley.CrossRefGoogle Scholar
[16]McDiarmid, C., Steger, A. and Welsh, D. J. A. (2005) Random planar graphs. J. Combin. Theory Ser. B 93 187–205.CrossRefGoogle Scholar
[17]Moon, J. W. (1968) On the maximum degree in a random tree. Michigan Math. J. 15 429–432.CrossRefGoogle Scholar
[18]Stanley, R. P. (1997) Enumerative Combinatorics, Vol. 1, Vol. 49 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge.CrossRefGoogle Scholar
[19]Walsh, T. R. S. (1982) Counting labelled three-connected and homeomorphically irreducible two-connected graphs. J. Combin. Theory Ser. B 32 1–11.CrossRefGoogle Scholar
[20]Weissl, A. C. (2007) Random graphs with structural constraints. PhD Thesis.Google Scholar