Algorithmica

, Volume 7, Issue 1–6, pp 521–554 | Cite as

A linear-time algorithm for finding an ambitus

  • B. Mishra
  • R. E. Tarjan
Article
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Abstract

We devise a linear-time algorithm for finding an ambitus ín an undirected graph. An ambitus is a cycle in a graph containing two distinguished vertices such that certain different groups of bridges (calledB itp-,B itQ-, andB itPQ-bridges) satisfy the property that a bridge in one group does not interlace with any bridge in the other groups. Thus, an ambitus allows the graph to be cut into pieces, where, in each piece, certain graph properties may be investigated independently and recursively, and then the pieces can be pasted together to yield information about these graph properties in the original graph. In order to achieve a good time-complexity for such an algorithm employing the divide-and-conquer paradigm, it is necessary to find an ambitus quickly. We also show that, using ambitus, linear-time algorithms can be devised for abiding-path-finding and nonseparating-induced-cycle-finding problems.

Key words

Abiding-path All-bidirectional-edges problem Ambitus Bridge Nonseparating induced cycle 
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References

  1. [1]
    J. Cheriyan and S. N. Maheswari. Finding Nonseparating Induced Cycles and Independent Spanning Trees in 3-Connected graphs.Journal of Algorithms, 9:507–537, 1988.MATHCrossRefMathSciNetGoogle Scholar
  2. [2]
    J. Hopcroft and R. Tarjan. Dividing a Graph into Triconnected Components.SIAM Journal of Computing, 2:135–158, September 1973.CrossRefMathSciNetGoogle Scholar
  3. [3]
    J. Hopcroft and R. Tarjan. Efficient Planarity Testing.Journal of the Association for Computing Machinery, 21:549–568, October 1974.MATHMathSciNetGoogle Scholar
  4. [4]
    S. V. Krishnan, C. Pandu Rangan, and S. Seshadri. A Simple Linear Algorithm for Finding the Ambitus in a Planar Graph. Technical Report, Department of Computer Science, Indian Institute of Technology, Madras, March 1988.Google Scholar
  5. [5]
    B. Mishra. An Efficient Algorithm To Find All “Bidirectional” Edges of an Undirected Graph.Proceedings of the 25th Annual Symposium on Foundations of Computer Science, pages 207–216, 1984.Google Scholar
  6. [6]
    B. Mishra. Some Graph Theoretic Issues in VLSI Design. Ph.D. thesis, Carnegie-Mellon University, September 1985.Google Scholar
  7. [7]
    T. Ohtsuki.The Two Disjoint Path Problem and Wire Routing Design. Graph Theory and Algorithms (Eds. N. Saito, T. Nishizeki). Springer-Verlag, Berlin, October 1980.Google Scholar
  8. [8]
    O. Ore.The Four-Color Problem. Academic Press, New York, 1967.MATHGoogle Scholar
  9. [9]
    H. Sachs.Einführung in die Theorie der endlichen Graphen. Teubner, Leipzig, 1972.MATHGoogle Scholar
  10. [10]
    R. Sundar. Finding the Abiding Path and Its Applications to Graph Algorithms. Technical Report, Department of Computer Science and Engineering, Indian Institute of Technology, Madras, June 1987.Google Scholar
  11. [11]
    R. Tarjan. Depth-First Search and Linear Graph Algorithms.SIAM Journal of Computing, 1(2): 146–160, June 1972.MATHCrossRefMathSciNetGoogle Scholar
  12. [12]
    W. T. Tutte.Connectivity in Graphs. University of Toronto Press, Toronto, 1966.MATHGoogle Scholar
  13. [13]
    W. T. Tutte. Bridges and Hamiltonian Circuits in Planar Graphs.Aequationes Mathematicae, 15:1–33, 1977.MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag New York Inc. 1992

Authors and Affiliations

  • B. Mishra
    • 1
  • R. E. Tarjan
    • 2
    • 3
  1. 1.Department of Computer Science, Courant Institute of Mathematical SciencesNew York UniversityNew YorkUSA
  2. 2.Department of Computer SciencePrinceton UniversityPrincetonUSA
  3. 3.NEC Research InstitutePrincetonUSA

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