On the Number of Connected Sets in Bounded Degree Graphs

  • Kustaa KangasEmail author
  • 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 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. 1.
    Alon, N.: Independent sets in regular graphs and sum-free subsets of finite groups. Isr. J. Math. 73, 247–256 (1991)CrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle 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)CrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Gaspers, S., Kratsch, D., Liedloff, M.: On independent sets and bicliques in graphs. Algorithmica 62(3–4), 637–658 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Gaspers, S., Mnich, M.: Feedback vertex sets in tournaments. J. Graph Theory 72(1), 72–89 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Kahn, J.: An entropy approach to the hard-core model on bipartite graphs. Combin. Probab. Comput. 10, 219–237 (2001)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Moon, J.W., Moser, L.: On cliques in graphs. Isr. J. Math. 3, 23–28 (1965)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Zhao, Y.: The number of independent sets in a regular graph. Combin. Probab. Comput. 19, 315–320 (2010)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Kustaa Kangas
    • 1
    Email author
  • 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

Personalised recommendations