Morphing Triangle Contact Representations of Triangulations
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Published:2023-03-15
Issue:3
Volume:70
Page:991-1024
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ISSN:0179-5376
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Container-title:Discrete & Computational Geometry
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language:en
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Short-container-title:Discrete Comput Geom
Author:
Angelini PatrizioORCID, Chaplick StevenORCID, Cornelsen SabineORCID, Da Lozzo GiordanoORCID, Roselli VincenzoORCID
Abstract
AbstractA 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 $${\mathcal {F}}$$
F
-contact representation of a plane graph G, vertices are realized by internally disjoint elements from a family $${\mathcal {F}}$$
F
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 $${\mathcal {F}}$$
F
-contact representations we insist that at each time step (continuously throughout the morph) we have an $${\mathcal {F}}$$
F
-contact representation. We focus on the case when $$\mathcal {F}$$
F
is the family of triangles in $$\mathbb {R}^2$$
R
2
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 $${\mathcal {O}}(n^2)$$
O
(
n
2
)
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.
Funder
deutsche forschungsgemeinschaft ministero dell’istruzione, dell’università e della ricerca horizon 2020 framework programme
Publisher
Springer Science and Business Media LLC
Subject
Computational Theory and Mathematics,Discrete Mathematics and Combinatorics,Geometry and Topology,Theoretical Computer Science
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