eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2016-06-10
41:1
41:15
10.4230/LIPIcs.SoCG.2016.41
article
The Planar Tree Packing Theorem
Geyer, Markus
Hoffmann, Michael
Kaufmann, Michael
Kusters, Vincent
Tóth, Csaba
Packing graphs is a combinatorial problem where several given graphs are being mapped into a common host graph such that every edge is used at most once. In the planar tree packing problem we are given two trees T1 and T2 on n vertices and have to find a planar graph on n vertices that is the edge-disjoint union of T1 and T2. A clear exception that must be made is the star which cannot be packed together with any other tree. But according to a conjecture of Garcia et al. from 1997 this is the only exception, and all other pairs of trees admit a planar packing. Previous results addressed various special cases, such as a tree and a spider tree, a tree and a caterpillar, two trees of diameter four, two isomorphic trees, and trees of maximum degree three. Here we settle the conjecture in the affirmative and prove its general form, thus making it the planar tree packing theorem. The proof is constructive and provides a polynomial time algorithm to obtain a packing for two given nonstar trees.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol051-socg2016/LIPIcs.SoCG.2016.41/LIPIcs.SoCG.2016.41.pdf
graph drawing
simultaneous embedding
planar graph
graph packin