On the Number of Connected Sets in Bounded Degree Graphs

  • Kustaa Kangas
  • Petteri Kaski
  • Mikko Koivisto
  • Janne H. Korhonen
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8747)

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 \(d=5\).

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Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Kustaa Kangas
    • 1
  • Petteri Kaski
    • 2
  • Mikko Koivisto
    • 1
  • Janne H. Korhonen
    • 1
  1. 1.Helsinki Institute for Information Technology HIIT, Department of Computer ScienceUniversity of HelsinkiHelsinkiFinland
  2. 2.Helsinki Institute for Information Technology HIIT, Department of Information and Computer ScienceAalto UniversityAaltoFinland

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