Advertisement

Acyclic Bidirected and Skew-Symmetric Graphs: Algorithms and Structure

  • Maxim A. Babenko
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3967)

Abstract

Bidirected graphs (a sort of nonstandard graphs introduced by Edmonds and Johnson) provide a natural generalization to the notions of directed and undirected graphs. By a weakly acyclic bidirected graph we mean such a graph having no simple cycles. We call a bidirected graph strongly acyclic if it has no cycles (even non-simple). We present (generalizing results of Gabow, Kaplan, and Tarjan) a modification of the depth-first search algorithm that checks (in linear time) if a given bidirected graph is weakly acyclic (in case of negative answer a simple cycle is constructed). We use the notion of skew-symmetric graphs (the latter give another, somewhat more convenient graph language which is essentially equivalent to the language of bidirected graphs). We also give structural results for the class of weakly acyclic bidirected and skew-symmetric graphs explaining how one can construct any such graph starting from strongly acyclic instances and, vice versa, how one can decompose a weakly acyclic graph into strongly acyclic “parts”. Finally, we extend acyclicity test to build (in linear time) such a decomposition.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Babenko, M.A., Karzanov, A.V.: Free multiflows in bidirected and skew-symmetric graphs (2005), Submitted to a special issue of DAMGoogle Scholar
  2. 2.
    Cormen, T., Leiserson, C., Rivest, R.: Introduction to Algorithms. MIT Press, Cambridge (1990)zbMATHGoogle Scholar
  3. 3.
    Edmonds, J., Johnson, E.L.: Matching, a well-solved class of integer linear programs. Combinatorial Structures and Their Applications, 89–92 (1970)Google Scholar
  4. 4.
    Gabow, H.N., Kaplan, H., Tarjan, R.E.: Unique maximum matching algorithms, pp. 70–78 (1999)Google Scholar
  5. 5.
    Gabow, H.N., Tarjan, R.E.: A linear-time algorithm for a special case of disjoint set union. J. Comp. and Syst. Sci. 30, 209–221 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Goldberg, A.V., Karzanov, A.V.: Path problems in skew-symmetric graphs. Combinatorica 16(3), 353–382 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Goldberg, A.V., Karzanov, A.V.: Maximum skew-symmetric flows and matchings. Mathematical Programming 100(3), 537–568 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Lovász, L., Plummer, M.D.: Matching Theory. North-Holland, NY (1986)zbMATHGoogle Scholar
  9. 9.
    Schrijver, A.: Combinatorial Optimization, vol. A. Springer, Berlin (2003)zbMATHGoogle Scholar
  10. 10.
    Tutte, W.T.: Antisymmetrical digraphs. Canadian J. Math. 19, 1101–1117 (1967)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Maxim A. Babenko
    • 1
  1. 1.Dept. of Mechanics and MathematicsMoscow State University, Vorob’yovy GoryMoscowRussia

Personalised recommendations