WADS 2013: Algorithms and Data Structures pp 353-364

# Planar Packing of Binary Trees

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

## Abstract

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 T1 and T2 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 Ti, 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.

## Preview

### References

1. 1.
Akiyama, J., Chvátal, V.: Packing paths perfectly. Discrete Mathematics 85(3), 247–255 (1990)
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)
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)
5. 5.
Eppstein, D.: Subgraph isomorphism in planar graphs and related problems. J. Graph Algorithms & Applications 3(3), 1–27 (1999)
6. 6.
Frank, A., Szigeti, Z.: A note on packing paths in planar graphs. Math. Program. 70(2), 201–209 (1995)
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)
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)
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.
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)
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)
15. 15.
Maheo, M., Saclé, J.-F., Woźniak, M.: Edge-disjoint placement of three trees. European J. Combin. 17(6), 543–563 (1996)
16. 16.
Mutzel, P., Odenthal, T., Scharbrodt, M.: The thickness of graphs: A survey. Graphs and Combinatorics 14(1), 59–73 (1998)
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)
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)
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)
21. 21.
Ullmann, J.R.: An algorithm for subgraph isomorphism. J. ACM 23(1), 31–42 (1976)

## 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