Partitions of Graphs into Trees

  • Therese Biedl
  • Franz J. Brandenburg
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4372)


In this paper, we study the k-tree partition problem which is a partition of the set of edges of a graph into k edge-disjoint trees. This problem occurs at several places with applications e.g. in network reliability and graph theory. In graph drawing there is the still unbeaten (n − 2) ×(n − 2) area planar straight line drawing of maximal planar graphs using Schnyder’s realizers [15], which are a 3-tree partition of the inner edges. Maximal planar bipartite graphs have a 2-tree partition, as shown by Ringel [14]. Here we give a different proof of this result with a linear time algorithm. The algorithm makes use of a new ordering which is of interest of its own. Then we establish the NP-hardness of the k-tree partition problem for general graphs and k ≥ 2. This parallels NP-hard partition problems for the vertices [3], but it contrasts the efficient computation of partitions into forests (also known as arboricity) by matroid techniques [7].


  1. 1.
    Biedl, T., Brandenburg, F.J.: Drawing planar bipartite graphs with small area. In: Proc. 17th Canadian Conference on Computational Geometry, CCCG’05, vol. 17, University of Windsor, Ontario, Canada, August 1-10 (2005)Google Scholar
  2. 2.
    Brandstädt, A., Le, V.B., Symczak, T.: The complexity of some problems related to graph 3-colorability. Disc. Appl. Math. 89(1), 59–73 (1998)CrossRefzbMATHGoogle Scholar
  3. 3.
    Broersma, H., Fomin, F.V., Kratochvil, J., Woeginger, G.J.: Planar graph coloring avoiding monochromatic subgraphs: Trees and paths make it difficult. Algorithmica 44, 343–361 (2006)CrossRefMathSciNetzbMATHGoogle Scholar
  4. 4.
    de Fraysseix, H., Ossona de Mendez, P., Pach, J.: Representation of planar graphs by segments. Intuitive Geometry 63, 109–117 (1991)Google Scholar
  5. 5.
    de Fraysseix, H., Pach, J., Pollack, R.: How to draw a planar graph on a grid. Combinatorica 10, 41–51 (1990)CrossRefMathSciNetzbMATHGoogle Scholar
  6. 6.
    Even, S., Tarjan, R.E.: Computing an st-numbering. Theoretical Computer Science 2, 436–441 (1976)CrossRefMathSciNetGoogle Scholar
  7. 7.
    Gabow, H.N., Westermann, H.H.: Forests, frames, and games: Algorithms for matroid sums and applications. Algorithmica 7, 465–497 (1992)CrossRefMathSciNetzbMATHGoogle Scholar
  8. 8.
    Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W.H. Freeman, San Francisco (1979)zbMATHGoogle Scholar
  9. 9.
    Kant, G.: Drawing planar graphs using the canonical ordering. Algorithmica 16, 4–32 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Lempel, A., Even, S., Cederbaum, I.: An algorithm for planarity testing of graphs. In: Theory of Graphs, International Symposium, Rome, 1966, pp. 215–232. Gordon and Breach, London (1967)Google Scholar
  11. 11.
    Nash-Williams, C.S.J.: Edge-disjoint spanning trees of finite graphs. J. London Math. Soc. 36, 445–450 (1961)CrossRefMathSciNetzbMATHGoogle Scholar
  12. 12.
    Nash-Williams, C.S.J.: Decomposition of finite graphs into forests. J. London Math. Soc. 39, 12 (1964)CrossRefMathSciNetzbMATHGoogle Scholar
  13. 13.
    Papadimitriou, C.H.: Computational Complexity. Addison-Wesley, Reading (1994)zbMATHGoogle Scholar
  14. 14.
    Ringel, G.: Two trees in maximal planar bipartite graphs. J. Graph Theory 17, 755–758 (1993)CrossRefMathSciNetzbMATHGoogle Scholar
  15. 15.
    Schnyder, W.: Embedding planar graphs on the grid. In: 1st Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 138–148. ACM Press, New York (1990)Google Scholar

Copyright information

© Springer Berlin Heidelberg 2007

Authors and Affiliations

  • Therese Biedl
    • 1
  • Franz J. Brandenburg
    • 2
  1. 1.School of Computer Science, University of Waterloo, N2L3G1Canada
  2. 2.Lehrstuhl für Informatik, Universität Passau, 94030 PassauGermany

Personalised recommendations