Geyer, Markus ;
Hoffmann, Michael ;
Kaufmann, Michael ;
Kusters, Vincent ;
Tóth, Csaba
The Planar Tree Packing Theorem
Abstract
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 edgedisjoint 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.
BibTeX  Entry
@InProceedings{geyer_et_al:LIPIcs:2016:5933,
author = {Markus Geyer and Michael Hoffmann and Michael Kaufmann and Vincent Kusters and Csaba T{\'o}th},
title = {{The Planar Tree Packing Theorem}},
booktitle = {32nd International Symposium on Computational Geometry (SoCG 2016)},
pages = {41:141:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959770095},
ISSN = {18688969},
year = {2016},
volume = {51},
editor = {S{\'a}ndor Fekete and Anna Lubiw},
publisher = {Schloss DagstuhlLeibnizZentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2016/5933},
URN = {urn:nbn:de:0030drops59337},
doi = {10.4230/LIPIcs.SoCG.2016.41},
annote = {Keywords: graph drawing, simultaneous embedding, planar graph, graph packin}
}
2016
Keywords: 

graph drawing, simultaneous embedding, planar graph, graph packin 
Seminar: 

32nd International Symposium on Computational Geometry (SoCG 2016)

Issue date: 

2016 
Date of publication: 

2016 