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Bounded degree spanning trees

Extended abstract
  • Artur Czumaj
  • Willy-B. Strothmann
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1284)

Abstract

Given a connected graph G, let a ΔT-spanning tree of G be a spanning tree of G of maximum degree bounded by ΔT. It is well known that for each ΔT ≥ 2 the problem of deciding whether a connected graph has a ΔT-spanning tree is NP-complete. In this paper we investigate this problem when additionally connectivity and maximum degree of the graph are given. A complete characterization of this problem for 2- and 3-connected graphs, for planar graphs, and for ΔT = 2 is provided.

Our first result is that given a biconnected graph of maximum degree 2ΔT - 2, we can find its ΔT-spanning tree in time O(m + n3/2). For graphs of higher connectivity we design a polynomial-time algorithm that finds a ΔT-spanning tree in any k-connected graph of maximum degree k(ΔT − 2) + 2. On the other hand, we prove that deciding whether a k-connected graph of maximum degree k(ΔT - 2) + 3 has a ΔT-spanning tree is NP-complete, provided k ≤ 3. For arbitrary k ≥ 3 we show that verifying whether a k-connected graph of maximum degree k(ΔT - 1) has a ΔT-spanning tree is NP-complete. In particular, we prove that the Hamiltonian path (cycle) problem is NP-complete for k-connected k-regular graphs, if k > 2. This extends the well known result for k = 3 and fully characterizes the case ΔT = 2.

For planar graphs it is NP-complete to decide whether a k-connected planar graph of maximum degree ΔG has a ΔT-spanning tree for k = 1 and Δg > ΔT ≥ 2, for k = 2 and ΔG > 2(ΔT -1) ≥ 2, and for k = 3 and ΔG > ΔT = 2. On the other hand, we show how to find in polynomial (linear or almost linear) time a ΔT-spanning tree for all other parameters of k, ΔG, and ΔT.

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References

  1. 1.
    D. Barnette. Trees in polyhedral graphs. Canadian J. Mathematics, 18:731–736, 1966.Google Scholar
  2. 2.
    D. Bauer, S. L. Hakimi, and E. F. Schmeichel. Recognizing tough graphs is NP-hard. Discrete Applied Mathematics, 28:191–195, 1990.Google Scholar
  3. 3.
    B. Bollobás. Extremal Graph Theory. Academic Press, London, 1978.Google Scholar
  4. 4.
    P. M. Camerini, G. Galgiati, and R. Maffioli. Complexity of spanning tree problems, I. European Journal of Operation Research, 5:346–352, 1980.Google Scholar
  5. 5.
    Y. Caro, I. Krasikov, and Y. Roditty. On the largest tree of a given maximum degree in a connected graph. Journal of Graph Theory, 15:7–13, 1991.Google Scholar
  6. 6.
    N. Chiba and T. Nishizeki. The Hamiltonian cycle problem is linear-time solvable for 4-connected planar graphs. Journal of Algorithms, 10:187–211, 1989.Google Scholar
  7. 7.
    M. Fairer and B. Raghavachari. Approximating the minimum-degree Steiner tree to within one of optimal. Journal of Algorithms, 17:409–423,1994. Also in ACM-SIAM SODA 1992.Google Scholar
  8. 8.
    M. R. Garey and D. S. Johnson. Computers and Intractability: A Guide to the Theory of NP-completeness. Freeman, New York, 1979.Google Scholar
  9. 9.
    M. R. Garey, D. S. Johnson, and R. E. Tarjan. The planar Hamiltonian circuit problem is NP-complete. SIAM Journal on Computing, 5(4):704–714, 1976.Google Scholar
  10. 10.
    M. X. Goemans and D. P. Williamson. A general approximation technique for constrained forest problems.SIAM Journal on Computing, 24(2):296–317, 1995.Google Scholar
  11. 11.
    H. Nagamochi and T. Ibaraki. A linear-time algorithm for finding a sparse k-connected spanning subgraph of a k-connected graph. Algorithmica, 7:583–596, 1992.Google Scholar
  12. 12.
    B. Jackson and T. D. Parsons. On r-regular r-connected non-Hamiltonian graphs. Bulletin of Australian Mathematics Society, 24:205–220, 1981.Google Scholar
  13. 13.
    D. S. Johnson. The NP-completeness column: An ongoing guide. Journal of Algorithms, 6:434–451, 1985.Google Scholar
  14. 14.
    S. Khuller and B. Raghavachari. Improved approximation algorithms for uniform connectivity problems. In Proceedings of the 27 ACM STOC, pp. 1–10, 1995.Google Scholar
  15. 15.
    S. Khuller, B. Raghavachari, and N. Young. Low degree spanning trees of small weight. SIAM Journal on Computing, 25(2):355–368, 1996.Google Scholar
  16. 16.
    S. Khuller and U. Vishkin. Biconnectivity approximations and graph carvings. Journal of the ACM, 41(2):214–235, 1994.Google Scholar
  17. 17.
    T. Lukovski and W.-B. Strothmann. Decremental biconnectivity on planar graphs. Manuscript, 1997.Google Scholar
  18. 18.
    G. H. J. Meredith. Regular n-valent n-connected nonHamiltonian non-n-edge-colorable graphs. Journal of Combinatorial Theory Series B, 14:55–60, 1973.Google Scholar
  19. 19.
    V. Neumann-Lara and E. Rivera-Campo. Spanning trees with bounded degrees. Combinatorica, 11(1):55–61, 1991.Google Scholar
  20. 20.
    C. H. Papadimitriou and M. Yannakakis. The complexity of restricted spanning tree problems. Journal of the ACM, 29(2):285–309, 1982.Google Scholar
  21. 21.
    M. Rauch. Improved data structures for fully dynamic biconnectivity. Full version. A preliminary version appeared in Proceedings of the 26th ACM STOC, 1994.Google Scholar
  22. 22.
    M. Rauch Henzinger and V King. Fully dynamic biconnectivity and transitive closure. In Proceedings of the 36th IEEE FOCS, pp. 664–673, 1995.Google Scholar
  23. 23.
    D.D. Sleator and R.E. Tarjan. A data structure for dynamic trees. Journal of Computer and System Sciences 26:362–391, 1983.Google Scholar
  24. 24.
    W.-B. Strothmann. Constructing 3-trees in 3-connected planar graphs in linear time. Manuscript, 1996.Google Scholar
  25. 25.
    S. Win. On a connection between the existence of k-trees and the toughness of a graph. Graphs and Combinatorics, 5:201–205, 1989.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1997

Authors and Affiliations

  • Artur Czumaj
    • 1
  • Willy-B. Strothmann
    • 1
  1. 1.Heinz Nixdorf Institute and Department of Computer ScienceUniversity of PaderbornPaderbornGermany

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