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)


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 \(d=5\).


Travel Salesman Problem Maximal Clique Boundary Vertex Neighborhood Graph Computer Search 


  1. 1.
    Alon, N.: Independent sets in regular graphs and sum-free subsets of finite groups. Isr. J. Math. 73, 247–256 (1991)CrossRefMATHGoogle Scholar
  2. 2.
    Binkele-Raible, D., Fernau, H., Gaspers, S., Liedloff, M.: Exact and parameterized algorithms for max internal spanning tree. Algorithmica 65(1), 95–128 (2013)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Björklund, A., Husfeldt, T., Kaski, P., Koivisto, M.: Computing the Tutte polynomial in vertex-exponential time. In: FOCS, pp. 677–686. IEEE Computer Society (2008)Google Scholar
  4. 4.
    Björklund, A., Husfeldt, T., Kaski, P., Koivisto, M.: Trimmed Moebius inversion and graphs of bounded degree. Theor. Comput. Syst. 47(3), 637–654 (2010)CrossRefMATHGoogle Scholar
  5. 5.
    Björklund, A., Husfeldt, T., Kaski, P., Koivisto, M.: The traveling salesman problem in bounded degree graphs. ACM Trans. Algorithms 8(2), 18:1–18:13 (2012)CrossRefGoogle Scholar
  6. 6.
    Bollobás, B.: The Art of Mathematics: Coffee Time in Memphis. Cambridge University Press (2006)Google Scholar
  7. 7.
    Chung, F., Graham, R., Frankl, P., Shearer, J.: Some intersection theorems for ordered sets and graphs. J. Comb. Theor. Ser. A 43(1), 23–37 (1986)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Fomin, F.V., Gaspers, S., Pyatkin, A.V., Razgon, I.: On the minimum feedback vertex set problem: exact and enumeration algorithms. Algorithmica 52(2), 293–307 (2008)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Fomin, F.V., Grandoni, F., Pyatkin, A.V., Stepanov, A.A.: Combinatorial bounds via measure and conquer: Bounding minimal dominating sets and applications. ACM Trans. Algorithms 5(1), 9:1–9:17 (2008)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Fomin, F.V., Villanger, Y.: Finding induced subgraphs via minimal triangulations. In: Marion, J.Y., Schwentick, T. (eds.) STACS. Volume 5 of LIPIcs., Schloss Dagstuhl - Leibniz-Zentrum für Informatik, pp. 383–394 (2010)Google Scholar
  11. 11.
    Fomin, F.V., Villanger, Y.: Treewidth computation and extremal combinatorics. Combinatorica 32(3), 289–308 (2012)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Galvin, D.: An upper bound for the number of independent sets in regular graphs. Discrete Math. 309(23–24), 6635–6640 (2009)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Gaspers, S., Kratsch, D., Liedloff, M.: On independent sets and bicliques in graphs. Algorithmica 62(3–4), 637–658 (2012)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Gaspers, S., Mnich, M.: Feedback vertex sets in tournaments. J. Graph Theory 72(1), 72–89 (2013)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Kahn, J.: An entropy approach to the hard-core model on bipartite graphs. Combin. Probab. Comput. 10, 219–237 (2001)MathSciNetMATHGoogle Scholar
  16. 16.
    Moon, J.W., Moser, L.: On cliques in graphs. Isr. J. Math. 3, 23–28 (1965)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Perrier, E., Imoto, S., Miyano, S.: Finding optimal Bayesian network given a super-structure. J. Mach. Learn. Res. 9, 2251–2286 (2008)MathSciNetMATHGoogle Scholar
  18. 18.
    Zhao, Y.: The number of independent sets in a regular graph. Combin. Probab. Comput. 19, 315–320 (2010)MathSciNetCrossRefMATHGoogle Scholar

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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