The Complexity of the Graph Embedding Problem

  • D. Archdeacon

Abstract

We investigate the computational complexity of determining if a graph G on v vertices embeds in a surface S. Robertson and Seymour have given an O(v3) decision algorithm for this embedding problem. We show here how the use the yes/no output from their algorithm to construct the embedding, that is, we self-reduce the search algorithm to the decision algorithm. We conclude that for each fixed surface S there exists an O(v10) algorithm for constructing an embedding or answering that no embedding exists.

Keywords

genus embedding computational complexity NP-complete 

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References

  1. [B]
    H.R. Brahana, Systems of circuits of two-dimensional manifolds, Ann. of Math. 30 (1923), 234–243.MathSciNetMATHGoogle Scholar
  2. [BFL]
    D.J. Brown, M.R. Fellows and M.A. Langston, Polynomial-time self-reducibility: theoretical motivations and practical results,Int. J. Computer Mathematics (to appear).Google Scholar
  3. [BHKY]
    J. Battle, F. Harary, Y. Kodama and J.W.T. Youngs, Additivity of the genus of a graph, Bull. Amer. Math. Soc. 68, 565–571.Google Scholar
  4. [D]
    R.A. Duke, The genus, regional number, and Betti number of a graph, Canad. J. Math. 18 (1966), 817–822.MathSciNetMATHCrossRefGoogle Scholar
  5. [FGM]
    M.L. Furst, L.L. Gross, and L.A. McGeoch, Finding a maximum-genus graph imbedding, preprint (1987).Google Scholar
  6. [FMR]
    I.S. Filotti, G.L. Miller and J. Reif, On determining the genus of a graph in O(v O(G)) steps., Proc. 11th Annual ACM Symp. Theory of Computing (1979), 27–37.Google Scholar
  7. [GJ]
    M.R. Garey and D.S. Johnson, “Computers and Intractability, a Guide to the Theory of NP-Completeness,” W.H. Freeman and Co., San Francisco, 1979.MATHGoogle Scholar
  8. [GT]
    J.L. Gross and T.W. Tucker, “Topological Graph Theory,” John Wiley & Sons, New York, 1987.MATHGoogle Scholar
  9. [HT]
    J.E. Hoperoft and R.E. Tarjan, Efficient planarity testing, J. Assoc. Comput. Math. 21 (1974), 549–568.CrossRefGoogle Scholar
  10. [MP]
    A. Meyer and M. Paterson, With what frequency are apparently intractable problems difficult, Technical Report, MIT (1979).Google Scholar
  11. [RS]
    N. Robertson and P.D. Seymour, Graph minors XIII: The disjoint paths problem,to appear.Google Scholar
  12. [T1]
    Carsten Thomassen, The graph genus problem is NP-complete,preprint.Google Scholar
  13. [T2]
    Carsten Thomassen, Embeddings of graphs with no short noncontractible cycles,preprint.Google Scholar
  14. [Y]
    J.W.T. Youngs, Minimal imbeddings and the genus of a graph, J. Math. Mech. 12 (1963), 303–315.MathSciNetMATHGoogle Scholar

Copyright information

© Physica-Verlag Heidelberg 1990

Authors and Affiliations

  • D. Archdeacon
    • 1
  1. 1.BurlingtonUSA

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