Cliques of a graph-variations on the Bron-Kerbosch algorithm

  • H. C. Johnston


This paper develops a family of algorithms that are variations on the Bron-Kerbosch algorithm for finding all the cliques of a simple undirected graph. The algorithms are developed in a stepwise manner, from a recursive algorithm for generating all combinations of zero or more objects chosen fromN objects. Experimental results are given.

Key words

Cliques combinatorial programming graph theory stepwise refinement tree search 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    E. A. Akkoyunlu, The enumeration of maximal cliques of large graphs,SIAM J. Computing 2:1–6 (March 1973).Google Scholar
  2. 2.
    A. P. Ambler, H. G. Barrow, C. M. Brown, R. M. Burstall, and R. J. Popplestone, A versatile computer controlled assembly system,Third International Joint Conference on Artifial Intelligence: Advance Papers (August 1973), pp. 298–303.Google Scholar
  3. 3.
    J. G. Augustson and J. Minker, An analysis of some graph theoretical cluster techniques,J. ACM 17: 571–588 (October 1970).Google Scholar
  4. 4.
    C. Bron and J. Kerbosch, “Algorithm 457: Finding all cliques of an undirected graph,”Commun. ACM 16:575–577 (September 1973).Google Scholar
  5. 5.
    C. Bron, J. Kerbosch, and H. J. Schell, “Finding Cliques in an Undirected Graph,” Technical Report, Technological University of Eindhoven, The Netherlands.Google Scholar
  6. 6.
    R. M. Burstall, “Tree searching methods with an application to a network design problem,”Mach. Intell. 1:65–85 (1967).Google Scholar
  7. 7.
    C.-M. Cheng, “Clustering by Clique Generation,” M.Sc. thesis, Department of Computer Science, University of Illinois, Urbana, Illinois (June 1974).Google Scholar
  8. 8.
    S. R. Das, “On a new approach for finding all the modified cut-sets in an incompatibility graph,”IEEE Trans. Comput. C-22:187–193 (February 1973).Google Scholar
  9. 9.
    J. W. Moon and L. Moser, “On cliques in graphs,”Isr. J. Math. 3:23–28 (1965).Google Scholar
  10. 10.
    G. D. Mulligan, “Algorithms for Finding Cliques of a graph,” Technical Report No. 41, Department of Computer Science, University of Toronto, Ontario, Canada (May 1972).Google Scholar
  11. 11.
    G. D. Mulligan and D. G. Corneil, “Corrections to Bierstone's algorithm for generating cliques,”J. ACM 19:244–247 (April 1972).Google Scholar
  12. 12.
    R. E. Osteen and J. T. Tou, “A clique-detection algorithm based on neighborhoods in graphs,”Int. J. Comput. Inf. Sci. 2:257–268 (1973).Google Scholar
  13. 13.
    K. A. Van Lehn, “SAIL User Manual,” Technical Report STAN-CS-73-373, Computer Science Department, Stanford University, California (July 1973).Google Scholar

Copyright information

© Plenum Publishing Corporation 1976

Authors and Affiliations

  • H. C. Johnston
    • 1
  1. 1.Department of Computer ScienceQueen's University of BelfastBelfastNorthern Ireland

Personalised recommendations