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DOI: 10.4230/LIPIcs.SoCG.2016.41

The Planar Tree Packing Theorem

Authors: Markus Geyer, Michael Hoffmann, Michael Kaufmann, Vincent Kusters, and Csaba Tóth

Published in: LIPIcs, Volume 51, 32nd International Symposium on Computational Geometry (SoCG 2016)

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 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.

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Markus Geyer, Michael Hoffmann, Michael Kaufmann, Vincent Kusters, and Csaba Tóth. The Planar Tree Packing Theorem. In 32nd International Symposium on Computational Geometry (SoCG 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 51, pp. 41:1-41:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{geyer_et_al:LIPIcs.SoCG.2016.41,
  author =	{Geyer, Markus and Hoffmann, Michael and Kaufmann, Michael and Kusters, Vincent and T\'{o}th, Csaba},
  title =	{{The Planar Tree Packing Theorem}},
  booktitle =	{32nd International Symposium on Computational Geometry (SoCG 2016)},
  pages =	{41:1--41:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-009-5},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{51},
  editor =	{Fekete, S\'{a}ndor and Lubiw, Anna},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2016.41},
  URN =		{urn:nbn:de:0030-drops-59337},
  doi =		{10.4230/LIPIcs.SoCG.2016.41},
  annote =	{Keywords: graph drawing, simultaneous embedding, planar graph, graph packin}
}

Document

DOI: 10.4230/DagSemProc.08191.6

08191 Working Group Summary – Visually Comparing a Set of Graphs

Authors: Mario Albrecht, Alejandro Estrella-Balderrama, Markus Geyer, Carsten Gutwenger, Karsten Klein, Oliver Kohlbacher, and Michael Schulz

Published in: Dagstuhl Seminar Proceedings, Volume 8191, Graph Drawing with Applications to Bioinformatics and Social Sciences (2008)

Abstract

We consider methods to visually compare graphs, more to focus on the differences of the graphs than on the similarities. Our two-level approach constructs a meaningful overview of the given graphs combined with a detailed view focusing on a local area of change. The actual layout of these graphs has to be evaluated depending on the specific type of biological network to be visualized in each case. We look into different variants and propose properties to be optimized in our visualizations.

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Mario Albrecht, Alejandro Estrella-Balderrama, Markus Geyer, Carsten Gutwenger, Karsten Klein, Oliver Kohlbacher, and Michael Schulz. 08191 Working Group Summary – Visually Comparing a Set of Graphs. In Graph Drawing with Applications to Bioinformatics and Social Sciences. Dagstuhl Seminar Proceedings, Volume 8191, pp. 1-6, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2008)

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@InProceedings{albrecht_et_al:DagSemProc.08191.6,
  author =	{Albrecht, Mario and Estrella-Balderrama, Alejandro and Geyer, Markus and Gutwenger, Carsten and Klein, Karsten and Kohlbacher, Oliver and Schulz, Michael},
  title =	{{08191 Working Group Summary – Visually Comparing a Set of Graphs}},
  booktitle =	{Graph Drawing with Applications to Bioinformatics and Social Sciences},
  pages =	{1--6},
  series =	{Dagstuhl Seminar Proceedings (DagSemProc)},
  ISSN =	{1862-4405},
  year =	{2008},
  volume =	{8191},
  editor =	{Stephen P. Borgatti and Stephen Kobourov and Oliver Kohlbacher and Petra Mutzel},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.08191.6},
  URN =		{urn:nbn:de:0030-drops-15536},
  doi =		{10.4230/DagSemProc.08191.6},
  annote =	{Keywords: Graph drawing, visual graph comparison}
}

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Documents authored by Geyer, Markus

The Planar Tree Packing Theorem

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08191 Working Group Summary – Visually Comparing a Set of Graphs

Abstract

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