2016/03/24 by Markus Geyer, Michael Hoffmann, Geyer, Markus +8
Computer Science · Engineering · Mathematics · #Advanced Graph Theory Research #Algorithms and Data Compression #Combinatorics #Computational Geometry (cs.CG) #Computer graphics (images) #Computer science #F.2.2 #FOS: Computer and information sciences #G.2.2 #Mathematics #Planar #Tree (set theory) #VLSI and FPGA Design Techniques #cs.CG
paper · pdf · doi:10.48550/arxiv.1603.07737
Full version of our SoCG 2016 paper
arxiv created 2016/03/24 · openalex publication_date 2016/03/24 · arxiv updated 2016/03/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 García 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.