Relatively Bridge-Addable Classes of Graphs

  • Colin McDiarmid
  • Kerstin Weller
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8392)


In recent years there has been a growing interest in random graphs sampled uniformly from a suitable structured class of (labelled) graphs, such as planar graphs. In particular, bridge-addable classes have received considerable attention. A class of graphs is called bridge-addable if for each graph in the class and each pair u and v of vertices in different components, the graph obtained by adding an edge joining u and v must also be in the class. The concept was introduced in 2005 by McDiarmid, Steger and Welsh, who showed that, for a random graph sampled uniformly from such a class, the probability that it is connected is at least 1/e.

In this extended abstract, we generalise this result to relatively bridge-addable classes of graphs, which are classes of graphs where some but not necessarily all of the possible bridges are allowed to be introduced. We also give a bound on the expected number of vertices not in the largest component. These results are related to the theory of expander graphs. Furthermore, we investigate whether these bounds are tight, and in particular give detailed results about random forests in the bipartite graph K n/2,n/2.


random graphs labelled graphs bridge-addable expander-graphs forests in Kn,n 


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  1. 1.
    Addario-Berry, L., McDiarmid, C., Reed, B.: Connectivity for Bridge-Addable Monotone Graph Classes. Combinatorics, Probability and Computing 21, 803–815 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Alon, N., Spencer, J.H.: The Probabilistic Method. John Wiley & Sons Inc. (2008)Google Scholar
  3. 3.
    Balister, P., Bollobás, B., Gerke, S.: Connectivity of addable graph classes. Journal of Combinatorial Theory, Series B 98, 577–584 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Bollobás, B.: Random Graphs. Cambridge University Press (2001)Google Scholar
  5. 5.
    Kang, M., Panagiotou, K.: On the connectivity of random graphs from addable classes. Journal of Combinatorial Theory, Series B 103, 306–312 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    McDiarmid, C.: Connectivity for random graphs from a weighted bridge-addable class. Electronic Journal of Combinatorics 19, Paper 53, 20 (2012)Google Scholar
  7. 7.
    McDiarmid, C.: Random graphs from a minor-closed class. Combinatorics, Probability and Computing 18, 583–599 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    McDiarmid, C., Steger, A., Welsh, D.J.A.: Random Graphs from Planar and Other Addable Classes. In: Topics in Discrete Mathematics. Algorithms and Combinatorics, vol. 26, pp. 231–246. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  9. 9.
    McDiarmid, C., Steger, A., Welsh, D.J.A.: Random planar graphs. Journal of Combinatorial Theory, Series B 93, 187–205 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    McDiarmid, C., Weller, K.: Connectivity for relatively bridge-addable classes of graphs (in preparation)Google Scholar
  11. 11.
    Moon, J.W.: Counting Labelled Trees, Canadian Mathematical Congress, Montreal, Quebec (1970)Google Scholar
  12. 12.
    Rényi, A.: Some remarks on the theory of trees. Magyar Tud. Akad. Mat. Kutató Int. Közl 4, 73–85 (1959)zbMATHGoogle Scholar
  13. 13.
    Scoins, H.I.: The number of trees with nodes of alternate parity. In: Proc. Cambridge Philos. Soc., vol. 58, pp. 12–16 (1962)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Colin McDiarmid
    • 1
  • Kerstin Weller
    • 2
  1. 1.Department of StatisticsUniversity of OxfordUnited Kingdom
  2. 2.Institut für Theoretische InformatikETH ZürichSwitzerland

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