Spanning Trees with Many Leaves in Regular Bipartite Graphs

  • Emanuele G. Fusco
  • Angelo Monti
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4835)

Abstract

Given a d-regular bipartite graph Gd, whose nodes are divided in black nodes and white nodes according to the partition, we consider the problem of computing the spanning tree of Gd with the maximum number of black leaves. We prove that the problem is NP hard for any fixed d ≥ 4 and we present a simple greedy algorithm that gives a constant approximation ratio for the problem. More precisely our algorithm can be used to get in linear time an approximation ratio of 2 − 2/(d − 1)2 for d ≥ 4. When applied to cubic bipartite graphs the algorithm only achieves a 2-approximation ratio. Hence we introduce a local optimization step that allows us to improve the approximation ratio for cubic bipartite graphs to 1.5.

Focusing on structural properties, the analysis of our algorithm proves a lower bound on lB(n,d), i.e., the minimum m such that every Gd with n black nodes has a spanning tree with at least m black leaves. In particular, for d = 3 we prove that lB(n,3) is exactly \(\left\lceil\frac{n}{3}\right\rceil +1\).

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References

  1. 1.
    Bodlaender, H.L.: On linear time minor tests and depth first search. In: Dehne, F., Santoro, N., Sack, J.-R. (eds.) WADS 1989. LNCS, vol. 382, pp. 577–590. Springer, Heidelberg (1989)Google Scholar
  2. 2.
    Bonsma, P.S., Brüggemann, T., Woeginger, G.J.: A faster fpt algorithm for finding spanning trees with many leaves. In: Rovan, B., Vojtáš, P. (eds.) MFCS 2003. LNCS, vol. 2747, pp. 259–268. Springer, Heidelberg (2003)Google Scholar
  3. 3.
    Caro, Y., West, D.B., Yuster, R.: Connected domination and spanning trees with many leaves. SIAM Journal on Discrete Mathematics 13(2), 202–211 (2000)CrossRefMathSciNetGoogle Scholar
  4. 4.
    Ding, G., Johnson, T., Seymour, P.: Spanning trees with many leaves. Journal of Graph Theory 37, 189–197 (2001)MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Fujie, T.: The maximum-leaf spanning tree problem: Formulations and facets. Networks 43(4), 212–223 (2004)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Galbiati, G., Maffioli, F., Morzenti, A.: A short note on the approximability of the maximum leaves spanning tree problem. Inf. Process. Lett. 52(1), 45–49 (1994)MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Garey, M.R., Johnson, D.S.: Computers and Intractability; A Guide to the Theory of NP-Completeness. W. H. Freeman & Co., New York (1979)MATHGoogle Scholar
  8. 8.
    Griggs, J.R., Kleitman, D.J., Shastri, A.: Spanning trees with many leaves in cubic graphs. Journal of Graph Theory 13(6), 669–695 (1989)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Griggs, J.R., Wu, M.: Spanning trees in graphs of minimum degree 4 or 5. Discrete Math. 104(2), 167–183 (1992)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Kleitman, D.J., West, D.B.: Spanning trees with many leaves. SIAM Journal on Discrete Mathematics 4(1), 99–106 (1991)MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Lemke, P.: The maximum leaf spanning tree problem for cubic graphs is np-complete. Technical Report IMA Preprint Series # 428, University of Minnesota (July 1988)Google Scholar
  12. 12.
    Li, P.C., Toulouse, M.: Variations of the maximum leaf spanning tree problem for bipartite graphs. Inf. Process. Lett. 97(4), 129–132 (2006)CrossRefMathSciNetGoogle Scholar
  13. 13.
    Lorys, K., Zwozniak, G.: Approximation algorithm for the maximum leaf spanning tree problem for cubic graphs. In: Möhring, R.H., Raman, R. (eds.) ESA 2002. LNCS, vol. 2461, pp. 686–697. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  14. 14.
    Lu, H.-I, Ravi, R.: The power of local optimizations: Approximation algorithms for maximum-leaf spanning tree (draft)*. Technical Report CS-96-05 (1996)Google Scholar
  15. 15.
    Lu, H.-I, Ravi, R.: Approximating maximum leaf spanning trees in almost linear time. J. Algorithms 29(1), 132–141 (1998)MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Rahman, M.S., Kaykobad, M.: Complexities of some interesting problems on spanning trees. Inf. Process. Lett. 94(2), 93–97 (2005)CrossRefMathSciNetGoogle Scholar
  17. 17.
    Solis-Oba, R.: 2-approximation algorithm for finding a spanning tree with maximum number of leaves. In: Bilardi, G., Pietracaprina, A., Italiano, G.F., Pucci, G. (eds.) ESA 1998. LNCS, vol. 1461, pp. 441–452. Springer, Heidelberg (1998)CrossRefGoogle Scholar
  18. 18.
    Storer, J.A.: Constructing full spanning trees for cubic graphs. Inf. Process. Lett. 13, 8–11 (1981)MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Emanuele G. Fusco
    • 1
  • Angelo Monti
    • 1
  1. 1.Dipartimento di Informatica, “Sapienza”, Università di Roma , via Salaria, 113-00198 RomeItaly

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