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Pruning Graphs with Digital Search Trees. Application to Distance Hereditary Graphs

  • Jean-Marc Lanlignel
  • Olivier Raynaud
  • Eric Thierry
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1770)

Abstract

Given a graph, removing pendant vertices (vertices with only one neighbor) and vertices that have a twin (another vertex that has the same neighbors) until it is not possible yields a reduced graph, called the “pruned graph”. In this paper, we present an algorithm which computes this “pruned graph” either in linear time or in linear space. In order to achieve these complexity bounds, we introduce a data structure based on digital search trees. Originally designed to store a family of sets and to test efficiently equalities of sets after the removal of some elements, this data structure finds interesting applications in graph algorithmics. For instance, the computation of the “pruned graph” provides a new and simply implementable algorithm for the recognition of distance-hereditary graphs, and we improve the complexity bounds for the complete bipartite cover problem on bipartite distance-hereditary graphs.

Keywords

graph algorithms distance-hereditary graphs sets digital search trees amortized complexity 

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References

  1. [AHU71]
    A. V. Aho, I. E. Hopcroft, and J. D. Ullman. The design and analysis of computer algorithms. Addison-Welsey, 1974, exercise 2.12 on page 71.Google Scholar
  2. [BM86]
    H. J. Bandelt and H. M. Mulder. Distance-hereditary graphs. J. Combin. Theory, Ser. B, 41:182–208, 1986.zbMATHCrossRefMathSciNetGoogle Scholar
  3. [Dam97]
    G. Damiand. Quelques propriétés des graphes distances héréditaires. Master’s thesis, Université de Montpellier II, LIRMM, 1997.Google Scholar
  4. [GJ79]
    M. R. Garey and D. S. Johnson. Computers and Intractability: A guide to the theory of NP-completeness. Freeman, 1979.Google Scholar
  5. [HM90]
    P. Hammer and F. Maffray. Completely separable graphs. Discrete Applied Mathematics, 27:85–99, 1990.CrossRefzbMATHMathSciNetGoogle Scholar
  6. [HMPV97]
    M. Habib, R. McConnell, C. Paul, and L. Viennot. LexBFS and partition refinement, with applications to transitive orientation, interval graph recognition and consecutive ones testing. Theor. Comp. Sci., 1999. to appear.Google Scholar
  7. [How77]
    E. Howorka. A characterization of distance-hereditary graphs. Quart. J. Math. Oxford, Ser. 2, 28:417–420, 1977.zbMATHCrossRefMathSciNetGoogle Scholar
  8. [Knu73]
    D. E. Knuth. The Art of Computer Programming: Sorting and Searching, volume 3. Addison-Wesley, 1973.Google Scholar
  9. [Mül96]
    H. Müller. On edge perfectness and classes of bipartite graphs. Discrete Mathematics, 149:159–187, 1996.zbMATHCrossRefMathSciNetGoogle Scholar
  10. RTL76]
    D. J. Rose, R. E. Tarjan, and G. S. Leuker. Algorithmic aspects of vertex elimination on graphs. SIAM J. of Computing, 5(2):266–283, June 1976.zbMATHCrossRefGoogle Scholar
  11. [WWW]
    A more detailed version of this article http://www.lirmm.fr/~thierry

Copyright information

© Springer-Verlag Berlin Heidelberg 2000

Authors and Affiliations

  • Jean-Marc Lanlignel
    • 1
  • Olivier Raynaud
    • 1
  • Eric Thierry
    • 1
  1. 1.LIRMMMontpellier Cedex 5France

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