A topological approach to evasiveness


The complexity of a digraph property is the number of entries of the vertex adjacency matrix of a digraph which must be examined in worst case to determine whether the graph has the property. Rivest and Vuillemin proved the result (conjectured by Aanderaa and Rosenberg) that every graph property that is monotone (preserved by addition of edges) and nontrivial (holds for some but not all graphs) has complexity Ω(v 2) wherev is the number of vertices. Karp conjectured that every such property is evasive, i.e., requires that every entry of the incidence matrix be examined. In this paper the truth of Karp’s conjecture is shown to follow from another conjecture concerning group actions on topological spaces. A special case of the conjecture is proved which is applied to prove Karp’s conjecture for the case of properties of graphs on a prime power number of vertices.

This is a preview of subscription content, access via your institution.


  1. [1]

    M. R. Best, P. van Emde Boas andH. W. Lenstra, Jr., A sharpened version of the Aanderaa-Rosenberg Conjecture,Report ZW 30/74, Mathematisch Centrum Amsterdam, 1974.

  2. [2]

    B. Bollobás, Complete subgraphs are elusive,J. Combinatorial Th. (B) 21 (1976), 1–7.

    MATH  Article  Google Scholar 

  3. [3]

    B. Bollobás,Extremal Graph Theory, Academic Press, 1978.

  4. [4]

    L. C. Glaser,Geometrical Combinatorial Topology, vol. 1, Van Nostrand, New York, 1970.

    Google Scholar 

  5. [5]

    G. H. Hardy andE. W. Wright,An Introduction to the Theory of Numbers, Clarendon Press, 1938.

  6. [6]

    Illies,to appear in Graph Theory Newsletter.

  7. [7]

    D. Kirkpatrick, Determining graph properties from matrix representations,in: Proceedings of 6 th SIGACT Conference, Seattle, (1974), ACM, 1975, 84–90.

  8. [8]

    D. J. Kleitman andD. J. Kwiatkowski, Further results on the Aanderaa—Rosenberg Conjecture,J. Comb. Th. B 28 (1980), 85–95.

    MATH  Article  MathSciNet  Google Scholar 

  9. [9]

    W. S. Massey,Algebraic Topology: An Introduction, Harcourt Brace Janovich, New York, 1967.

    Google Scholar 

  10. [10]

    E. C. Milner andD. J. A. Welsh, On the computational complexity of graph theoretical properties,in: Proceedings 5 th British Columbia Conf. on Combinatorics (C. St. J. A. Nash-Williams and J. Sheehan, Eds.), 1975, 471–487.

  11. [11]

    R. Oliver, Fixed-point sets of group actions on finite cyclic complexes,Comment. Math. Helv. 50 (1975), 155–177.

    MATH  Article  MathSciNet  Google Scholar 

  12. [12]

    R. Rivest andS. Vuillemin, On recognizing graph properties from adjacency matrices,Theor. Comp. Sci. 3 (1978), 371–384.

    Article  MathSciNet  Google Scholar 

  13. [13]

    R. Rivest andS. Vuillemin, A generalization and proof of the Aanderaa-Rosenberg conjecture,in: Proceedings of 7 th SIGACT Conference, Albuquerque, (1975), ACM, 1976.

  14. [14]

    A. L. Rosenberg, On the time required to recognize properties of graphs: A problem.SIGACT News 5 (4) (1973), 15–16.

    Article  Google Scholar 

  15. [15]

    P. A. Smith, Fixed point theorems for periodic transformations,Amer. J. of Math. 63 (1941), 1–8.

    MATH  Article  Google Scholar 

  16. [16]

    E. H. Spanier,Algebraic Topology, McGraw-Hill, New York, 1966.

    Google Scholar 

  17. [17]

    A. H. Wallace,Algebraic Topology, Benjamin, New York, 1970.

    Google Scholar 

Download references

Author information



Additional information

Supported in part by an NSF postdoctoral fellowship

Supported in part by NSF under grant No. MCS-8102248

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Kahn, J., Saks, M. & Sturtevant, D. A topological approach to evasiveness. Combinatorica 4, 297–306 (1984). https://doi.org/10.1007/BF02579140

Download citation

AMS subject classification (1980)

  • 05 C 99
  • 55 U 05