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)

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.

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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

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