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On linear time minor tests and depth first search

  • Hans L. Bodlaender
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 382)

Abstract

Recent results on graph minors make it desirable to have efficient algorithms, that for a fixed set of graphs {H1, ..., H c }, test whether a given graph G contains at least one graph H i as a minor. In this paper we show the following result: if at least one graph H i is a minor of a 2 × k grid graph, and at least one graph H i is a minor of a circus graph, then one can test in \(\mathcal{O}\)(n) time whether a given graph G contains at least one graph H∈{H1, ..., H c } as a minor. This result generalizes a result of Fellows and Langston. The algorithm is based on depth first search and on dynamic programming on graphs with bounded treewidth. As a corollary, it follows that the MAXIMUM LEAF SPANNING TREE problem can be solved in linear time for fixed k. We also discuss that with small modifications, an algorithm of Fellows and Langston can be modified to an algorithm that finds in \(\mathcal{O}\)(k!2 k n) time a cycle (or path) in a given graph with length≥k if it exists.

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References

  1. [1]
    S. Arnborg. Efficient algorithms for combinatorial problems on graphs with bounded decomposability — A survey. BIT, 25:2–23, 1985.Google Scholar
  2. [2]
    S. Arnborg, J. Lagergren, and D. Seese. Problems easy for tree-decomposable graphs (extended abstract). In Proc. 15'th ICALP, pages 38–51. Springer Verlag, Lect. Notes in Comp. Sc. 317, 1988.Google Scholar
  3. [3]
    S. Arnborg and A. Proskurowski. Linear time algorithms for NP-hard problems on graphs embedded in k-trees. Trita-na-8404, Dept. of Num. Anal. and Comp. Sci., Royal Institute of Technology, Stockholm, Sweden, 1984.Google Scholar
  4. [4]
    H. L. Bodlaender. NC-algorithms for graphs with small treewidth. In Proc. Workshop on Graph-Theoretic Concepts in Computer Science WG'88, pages 1–10. Springer Verlag, LNCS 344, 1988.Google Scholar
  5. [5]
    B. Courcelle. The monadic second-order logic of graphs I: Recognizable sets of finite graphs. Technical Report I-8837, Dept. Comp. Sc, Univ. Bordeaux 1, 1988.Google Scholar
  6. [6]
    B. Courcelle. The monadic second-order logic of graphs III: Treewidth, forbidden minors and complexity issues. Manuscript, 1988.Google Scholar
  7. [7]
    P. Erdös and G. Szekeres. A combinatorial problem in geometry. Compos. Math., 2:464–470, 1935.Google Scholar
  8. [8]
    M. R. Fellows, D. K. Friesen, and M. A. Langston. On finding optimal and near-optimal lineal spanning trees. Algorithmica, 3:549–560, 1988.Google Scholar
  9. [9]
    M. R. Fellows and M. A. Langston. Fast search algorithms for graph layout permutation problems. Technical Report CS-88-189, Dept. of Comp. Sc., Washington State Univ., 1988.Google Scholar
  10. [10]
    M. R. Fellows and M. A. Langston. On seach, decision and the efficiency of polynomial-time algorithms. Technical Report CS-88-190, Dept. of Comp. Sc., Washington State Univ., 1988. To appear in Proc. STOC '89.Google Scholar
  11. [11]
    M. R. Fellows and M. A. Langston. On well-partial-order theory and its application to combinatorial problems of VLSI design. Technical Report CS-88-188, Dept. of Comp. Sc., Washington State Univ., 1988.Google Scholar
  12. [12]
    M. R. Garey and D. S. Johnson. Computers and Intractability, A Guide to the Theory of NP-Completeness. W.H. Freeman and Company, New York, 1979.Google Scholar
  13. [13]
    J. E. Hopcroft and R. E. Tarjan. Efficient planarity testing. J. ACM, 21:549–568, 1974.CrossRefGoogle Scholar
  14. [14]
    C. Lautemann. Efficient algorithms on context-free graph languages. In Proc. 15'th ICALP, pages 362–378. Springer Verlag, Lect. Notes in Comp. Sc. 317, 1988.Google Scholar
  15. [15]
    N. Megiddo, S. L. Hakimi, M. R. Garey, D. S. Johnson, and C. H. Papadimitriou. The complexity of searching a graph. J. ACM, 35:18–44, 1988.CrossRefGoogle Scholar
  16. [16]
    K. Mehlhorn. Data Structures and Algorithms 2. Graph Algorithms and NP-Completeness. Springer Verlag, Berlin, 1984.Google Scholar
  17. [17]
    B. Monien. How to find long paths efficiently. Annals of Disc. Math., 25:239–254, 1985.Google Scholar
  18. [18]
    N. Robertson and P. Seymour. Graph minors. XVI. Wagner's conjecture. To appear.Google Scholar
  19. [19]
    N. Robertson and P. Seymour. Graph minors. II. Algorithmic aspects of tree-width. J. of Algorithms, 7:309–322, 1986.Google Scholar
  20. [20]
    N. Robertson and P. Seymour. Graph minors. XIII. The disjoint paths problem. Manuscript., 1986.Google Scholar
  21. [21]
    P. Scheffler. Linear-time algorithms for NP-complete problems restricted to partial k-trees. Report R-MATH-03/87, Karl-Weierstrass-Institut Für Mathematik, Berlin, GDR, 1987.Google Scholar
  22. [22]
    T. Wimer. Linear algorithms on k-terminal graphs. PhD thesis, Dept. of Computer Science, Clemson University, 1987.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1989

Authors and Affiliations

  • Hans L. Bodlaender
    • 1
  1. 1.Department of Computer ScienceUniversity of UtrechtUtrechtthe Netherlands

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