Planar Packing of Binary Trees

  • Markus Geyer
  • Michael Hoffmann
  • Michael Kaufmann
  • Vincent Kusters
  • Csaba D. Tóth
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8037)


In the graph packing problem we are given several graphs and have to map them into a single host graph G such that each edge of G is used at most once. Much research has been devoted to the packing of trees, especially to the case where the host graph must be planar. More formally, the problem is: Given any two trees T 1 and T 2 on n vertices, we want a simple planar graph G on n vertices such that the edges of G can be colored with two colors and the subgraph induced by the edges colored i is isomorphic to T i , for i ∈ {1,2}.

A clear exception that must be made is the star tree which cannot be packed together with any other tree. But a popular hypothesis states that this is the only exception, and all other pairs of trees admit a planar packing. Previous proof attempts lead to very limited results only, which include a tree and a spider tree, a tree and a caterpillar, two trees of diameter four and two isomorphic trees.

We make a step forward and prove the hypothesis for any two binary trees. The proof is algorithmic and yields a linear time algorithm to compute a plane packing, that is, a suitable two-edge-colored host graph along with a planar embedding for it. In addition we can also guarantee several nice geometric properties for the embedding: vertices are embedded equidistantly on the x-axis and edges are embedded as semi-circles.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Akiyama, J., Chvátal, V.: Packing paths perfectly. Discrete Mathematics 85(3), 247–255 (1990)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Braß, P., Cenek, E., Duncan, C.A., Efrat, A., Erten, C., Ismailescu, D., Kobourov, S.G., Lubiw, A., Mitchell, J.S.B.: On simultaneous planar graph embeddings. Comput. Geom. 36(2), 117–130 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Caro, Y., Yuster, R.: Packing graphs: The packing problem solved. Electr. J. Comb. 4(1) (1997)Google Scholar
  4. 4.
    Eppstein, D.: Arboricity and bipartite subgraph listing algorithms. Information Processing Letters 51(4), 207–211 (1994)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Eppstein, D.: Subgraph isomorphism in planar graphs and related problems. J. Graph Algorithms & Applications 3(3), 1–27 (1999)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Frank, A., Szigeti, Z.: A note on packing paths in planar graphs. Math. Program. 70(2), 201–209 (1995)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Frati, F.: Embedding graphs simultaneously with fixed edges. In: Kaufmann, M., Wagner, D. (eds.) GD 2006. LNCS, vol. 4372, pp. 108–113. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  8. 8.
    Frati, F.: Planar packing of diameter-four trees. In: 21st Canadian Conference on Computational Geometry (CCCG 2009), pp. 95–98 (2009)Google Scholar
  9. 9.
    Frati, F., Geyer, M., Kaufmann, M.: Planar packings of trees and spider trees. Information Processing Letters 109(6), 301–307 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    García, A., Hernando, C., Hurtado, F., Noy, M., Tejel, J.: Packing trees into planar graphs. J. Graph Theory, 172–181 (2002)Google Scholar
  11. 11.
    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.zbMATHGoogle Scholar
  12. 12.
    Geyer, M., Kaufmann, M., Vrt’o, I.: Two trees which are self–intersecting when drawn simultaneously. In: Healy, P., Nikolov, N.S. (eds.) GD 2005. LNCS, vol. 3843, pp. 201–210. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  13. 13.
    Gonçalves, D.: Edge partition of planar graphs into two outerplanar graphs. In: Proc. 37th Annu. ACM Sympos. Theory Comput., pp. 504–512 (2005)Google Scholar
  14. 14.
    Hedetniemi, S.M., Hedetniemi, S.T., Slater, P.J.: A note on packing two trees into \(K\sb{N}\). Ars Combin. 11, 149–153 (1981)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Maheo, M., Saclé, J.-F., Woźniak, M.: Edge-disjoint placement of three trees. European J. Combin. 17(6), 543–563 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Mutzel, P., Odenthal, T., Scharbrodt, M.: The thickness of graphs: A survey. Graphs and Combinatorics 14(1), 59–73 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Nash-Williams, C.S.J.A.: Edge-Disjoint Spanning Trees of Finite Graphs. Journal of the London Mathematical Society-second Series s1-36, 445–450 (1961)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Oda, Y., Ota, K.: Tight planar packings of two trees. In: European Workshop on Computational Geometry, pp. 215–216 (2006)Google Scholar
  19. 19.
    Schnyder, W.: Planar graphs and poset dimension. Order 5, 323–343 (1989)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Tutte, W.T.: On the problem of decomposing a graph into n connected factors. Journal of the London Mathematical Society s1-36(1), 221–230 (1961)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Ullmann, J.R.: An algorithm for subgraph isomorphism. J. ACM 23(1), 31–42 (1976)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Markus Geyer
    • 1
  • Michael Hoffmann
    • 2
  • Michael Kaufmann
    • 1
  • Vincent Kusters
    • 2
  • Csaba D. Tóth
    • 3
    • 4
  1. 1.Wilhelm-Schickard-Institut für InformatikUniversität TübingenGermany
  2. 2.Institute of Theoretical Computer ScienceETH ZürichSwitzerland
  3. 3.California State University NorthridgeUSA
  4. 4.University of CalgaryCanada

Personalised recommendations