Abstract
It is known that graphs on n vertices with minimum degree at least 3 have spanning trees with at least n/4 + 2 leaves and that this can be improved to (n + 4)/3 for cubic graphs without the diamond K 4 − e as a subgraph. We generalize the second result by proving that every graph with minimum degree at least 3, without diamonds and certain subgraphs called blossoms, has a spanning tree with at least (n + 4)/3 leaves. We show that it is necessary to exclude blossoms in order to obtain a bound of the form n/3 + c.
We use the new bound to obtain a simple FPT algorithm, which decides in O(m) + O *(6.75k) time whether a graph of size m has a spanning tree with at least k leaves. This improves the best known time complexity for Max-Leaves Spanning Tree.
Access provided by Autonomous University of Puebla. Download to read the full chapter text
Chapter PDF
Similar content being viewed by others
Keywords
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
References
Bodlaender, H.L.: On linear time minor tests with depth-first search. J. Algorithms 14(1), 1–23 (1993)
Bonsma, P.S.: Sparse cuts, matching-cuts and leafy trees in graphs. PhD thesis, University of Twente, Enschede, the Netherlands (2006), http://purl.org/utwente/57117
Bonsma, P.S., Brueggemann, 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)
Correa, J.R., Fernandes, C.G., Matamala, M., Wakabayashi, Y.: A 5/3-approximation for finding spanning trees with many leaves in cubic graphs. In: WAOA 2007 (to appear, 2007)
Estivill-Castro, V., Fellows, M.R., Langston, M.A., Rosamond, F.A.: FPT is P-time extremal structure I. In: ACiD 2005. Texts in algorithmics, vol. 4, pp. 1–41. King’s College Publications
Flum, J., Grohe, M.: Parameterized complexity theory. Springer, Berlin (2006)
Griggs, J.R., Kleitman, D.J., Shastri, A.: Spanning trees with many leaves in cubic graphs. J. Graph Theory 13(6), 669–695 (1989)
Kleitman, D.J., West, D.B.: Spanning trees with many leaves. SIAM J. Discrete Math. 4(1), 99–106 (1991)
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)
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 2008 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Bonsma, P., Zickfeld, F. (2008). Spanning Trees with Many Leaves in Graphs without Diamonds and Blossoms. In: Laber, E.S., Bornstein, C., Nogueira, L.T., Faria, L. (eds) LATIN 2008: Theoretical Informatics. LATIN 2008. Lecture Notes in Computer Science, vol 4957. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-78773-0_46
Download citation
DOI: https://doi.org/10.1007/978-3-540-78773-0_46
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-78772-3
Online ISBN: 978-3-540-78773-0
eBook Packages: Computer ScienceComputer Science (R0)