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Morphing Triangle Contact Representations of Triangulations

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Datum

2023

Autor:innen

Angelini, Patrizio
Chaplick, Steven
Da Lozzo, Giordano
Roselli, Vincenzo

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Deutsche Forschungsgemeinschaft (DFG): 50974019

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Core Facility der Universität Konstanz

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Published

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Discrete & Computational Geometry. Springer. 2023, 70(3), pp. 991-1024. ISSN 0179-5376. eISSN 1432-0444. Available under: doi: 10.1007/s00454-022-00475-9

Zusammenfassung

A morph is a continuous transformation between two representations of a graph. We consider the problem of morphing between contact representations of a plane graph. In an -contact representation of a plane graph G, 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 G. In a morph between two -contact representations we insist that at each time step (continuously throughout the morph) we have an -contact representation. We focus on the case when is the family of triangles in 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. Moreover, they naturally correspond to 3-orientations. Thus, they provide a natural case to study regarding morphs of contact representations of plane graphs. We characterize the pairs of RT-representations admitting a morph between each other via the respective 3-orientations. Our characterization leads to a polynomial-time algorithm to decide whether there is a morph between two RT-representations of an n-vertex plane triangulation, and, if so, computes a morph with steps. Each of these steps is a linear morph moving the endpoints of each triangle at constant speed along straight-line trajectories. Our characterization also implies 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.

Zusammenfassung in einer weiteren Sprache

Fachgebiet (DDC)
510 Mathematik

Schlagwörter

Planar graphs, Graph drawing, Morphing, Contact representation

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ISO 690ANGELINI, Patrizio, Steven CHAPLICK, Sabine CORNELSEN, Giordano DA LOZZO, Vincenzo ROSELLI, 2023. Morphing Triangle Contact Representations of Triangulations. In: Discrete & Computational Geometry. Springer. 2023, 70(3), pp. 991-1024. ISSN 0179-5376. eISSN 1432-0444. Available under: doi: 10.1007/s00454-022-00475-9
BibTex
@article{Angelini2023-03-15Morph-66489,
  year={2023},
  doi={10.1007/s00454-022-00475-9},
  title={Morphing Triangle Contact Representations of Triangulations},
  number={3},
  volume={70},
  issn={0179-5376},
  journal={Discrete & Computational Geometry},
  pages={991--1024},
  author={Angelini, Patrizio and Chaplick, Steven and Cornelsen, Sabine and Da Lozzo, Giordano and Roselli, Vincenzo},
  note={DFG - Project-ID 50974019 - TRR 161 (B06) (Cornelsen)}
}
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DFG - Project-ID 50974019 - TRR 161 (B06) (Cornelsen)
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