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Morphing Contact Representations of Graphs

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

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ACM Subject Classification
  • Theory of computation → Computational geometry
  • Mathematics of computing → Graph algorithms
Keywords
  • Contact representations
  • Triangulations
  • Planar morphs
  • Schnyder woods

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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.

Cite As Get BibTex

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) https://doi.org/10.4230/LIPIcs.SoCG.2019.10

Author Details

Patrizio Angelini
  • University of Tübingen, Tübingen, Germany
Steven Chaplick
  • University of Würzburg, Würzburg, Germany
Sabine Cornelsen
  • University of Konstanz, Konstanz, Germany
Giordano Da Lozzo
  • Roma Tre University, Rome, Italy
Vincenzo Roselli
  • Roma Tre University, Rome, Italy

Funding

Work supported by DFG grant Ka812/17-1 and by MIUR-DAAD Joint Mobility Program n.57397196 (Angelini), by DFG grant WO$ $758/11-1 (Chaplick), and by MIUR Project "MODE" under PRIN 20157EFM5C, by MIUR Project "AHeAD" under PRIN 20174LF3T8, by MIUR-DAAD JMP N° 34120, and by H2020-MSCA-RISE project 734922 – "CONNECT" (Da Lozzo and Roselli).

Acknowledgements

This work began at the Graph and Network Visualization Workshop (GNV’18) in Heiligkreuztal. We thank S. Felsner, N. Heinsohn, and A. Lubiw for interesting discussions.

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