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On the Number of Connected Sets in Bounded Degree Graphs

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Graph-Theoretic Concepts in Computer Science (WG 2014)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 8747))

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Abstract

A set of vertices in a graph is connected if the set induces a connected subgraph. Using Shearer’s entropy lemma, we show that the number of connected sets in an \(n\)-vertex graph with maximum vertex degree \(d\) is \(O(1.9351^n)\) for \(d=3\), \(O(1.9812^n)\) for \(d=4\), and \(O(1.9940^n)\) for \(d=5\). Dually, we construct infinite families of generalized ladder graphs whose number of connected sets is bounded from below by \(\varOmega (1.5537^n)\) for \(d=3\), \(\varOmega (1.6180^n)\) for \(d=4\), and \(\varOmega (1.7320^n)\) for \(d=5\).

K.K., M.K., and J.K. supported by the Academy of Finland, grants 125637, 218153, and 255675. P.K. supported by the Academy of Finland, grants 252083 and 256287.

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Notes

  1. 1.

    Our implementation is available at http://www.cs.helsinki.fi/u/jwkangas/consets/.

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Authors and Affiliations

  1. Helsinki Institute for Information Technology HIIT, Department of Computer Science, University of Helsinki, P.O. Box 68, 00014, Helsinki, Finland

    Kustaa Kangas, Mikko Koivisto & Janne H. Korhonen

  2. Helsinki Institute for Information Technology HIIT, Department of Information and Computer Science, Aalto University, P.O. Box 15400, 00076, Aalto, Finland

    Petteri Kaski

Authors
  1. Kustaa Kangas
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  2. Petteri Kaski
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  3. Mikko Koivisto
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  4. Janne H. Korhonen
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Correspondence to Kustaa Kangas .

Editor information

Editors and Affiliations

  1. LITA, Université de Lorraine, Metz Cedex 01, France

    Dieter Kratsch

  2. Université d'Orléans, Orléans, France

    Ioan Todinca

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Kangas, K., Kaski, P., Koivisto, M., Korhonen, J.H. (2014). On the Number of Connected Sets in Bounded Degree Graphs. In: Kratsch, D., Todinca, I. (eds) Graph-Theoretic Concepts in Computer Science. WG 2014. Lecture Notes in Computer Science, vol 8747. Springer, Cham. https://doi.org/10.1007/978-3-319-12340-0_28

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  • DOI: https://doi.org/10.1007/978-3-319-12340-0_28

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