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DOI: 10.4230/LIPIcs.SoCG.2019.10

Morphing Contact Representations of Graphs

Authors: Patrizio Angelini, Steven Chaplick, Sabine Cornelsen, Giordano Da Lozzo, and Vincenzo Roselli

Published in: LIPIcs, Volume 129, 35th International Symposium on Computational Geometry (SoCG 2019)

Abstract

We consider the problem of morphing between contact representations of a plane graph. In a contact representation of a plane graph, vertices are realized by internally disjoint elements from a family of connected geometric objects. Two such elements touch if and only if their corresponding vertices are adjacent. These touchings also induce the same embedding as in the graph. In a morph between two contact representations we insist that at each time step (continuously throughout the morph) we have a contact representation of the same type. We focus on the case when the geometric objects are triangles that are the lower-right half of axis-parallel rectangles. Such RT-representations exist for every plane graph and right triangles are one of the simplest families of shapes supporting this property. Thus, they provide a natural case to study regarding morphs of contact representations of plane graphs. We study piecewise linear morphs, where each step is a linear morph moving the endpoints of each triangle at constant speed along straight-line trajectories. We provide a polynomial-time algorithm that decides whether there is a piecewise linear morph between two RT-representations of a plane triangulation, and, if so, computes a morph with a quadratic number of linear morphs. As a direct consequence, we obtain that for 4-connected plane triangulations there is a morph between every pair of RT-representations where the "top-most" triangle in both representations corresponds to the same vertex. This shows that the realization space of such RT-representations of any 4-connected plane triangulation forms a connected set.

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Patrizio Angelini, Steven Chaplick, Sabine Cornelsen, Giordano Da Lozzo, and Vincenzo Roselli. Morphing Contact Representations of Graphs. In 35th International Symposium on Computational Geometry (SoCG 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 129, pp. 10:1-10:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{angelini_et_al:LIPIcs.SoCG.2019.10,
  author =	{Angelini, Patrizio and Chaplick, Steven and Cornelsen, Sabine and Da Lozzo, Giordano and Roselli, Vincenzo},
  title =	{{Morphing Contact Representations of Graphs}},
  booktitle =	{35th International Symposium on Computational Geometry (SoCG 2019)},
  pages =	{10:1--10:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-104-7},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{129},
  editor =	{Barequet, Gill and Wang, Yusu},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2019.10},
  URN =		{urn:nbn:de:0030-drops-104145},
  doi =		{10.4230/LIPIcs.SoCG.2019.10},
  annote =	{Keywords: Contact representations, Triangulations, Planar morphs, Schnyder woods}
}

Document

DOI: 10.4230/LIPIcs.SOCG.2015.126

Optimal Morphs of Convex Drawings

Authors: Patrizio Angelini, Giordano Da Lozzo, Fabrizio Frati, Anna Lubiw, Maurizio Patrignani, and Vincenzo Roselli

Published in: LIPIcs, Volume 34, 31st International Symposium on Computational Geometry (SoCG 2015)

Abstract

We give an algorithm to compute a morph between any two convex drawings of the same plane graph. The morph preserves the convexity of the drawing at any time instant and moves each vertex along a piecewise linear curve with linear complexity. The linear bound is asymptotically optimal in the worst case.

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Patrizio Angelini, Giordano Da Lozzo, Fabrizio Frati, Anna Lubiw, Maurizio Patrignani, and Vincenzo Roselli. Optimal Morphs of Convex Drawings. In 31st International Symposium on Computational Geometry (SoCG 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 34, pp. 126-140, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{angelini_et_al:LIPIcs.SOCG.2015.126,
  author =	{Angelini, Patrizio and Da Lozzo, Giordano and Frati, Fabrizio and Lubiw, Anna and Patrignani, Maurizio and Roselli, Vincenzo},
  title =	{{Optimal Morphs of Convex Drawings}},
  booktitle =	{31st International Symposium on Computational Geometry (SoCG 2015)},
  pages =	{126--140},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-83-5},
  ISSN =	{1868-8969},
  year =	{2015},
  volume =	{34},
  editor =	{Arge, Lars and Pach, J\'{a}nos},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SOCG.2015.126},
  URN =		{urn:nbn:de:0030-drops-51333},
  doi =		{10.4230/LIPIcs.SOCG.2015.126},
  annote =	{Keywords: Convex Drawings, Planar Graphs, Morphing, Geometric Representations}
}

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Documents authored by Roselli, Vincenzo

Morphing Contact Representations of Graphs

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Optimal Morphs of Convex Drawings

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