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 F-contact representation of a plane graph G, vertices are realized by internally disjoint elements from a family 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&nbsp;G. In a morph between two F-contact representations we insist that at each time step (continuously throughout the morph) we have an F-contact representation. We focus on the case when F is the family of triangles in R2 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 O(n2) 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.

Angelini, P., Chaplick, S., Cornelsen, S., Da Lozzo, G., Roselli, V. (2023). Morphing Triangle Contact Representations of Triangulations. DISCRETE & COMPUTATIONAL GEOMETRY [10.1007/s00454-022-00475-9].

### Morphing Triangle Contact Representations of Triangulations

#### Abstract

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 F-contact representation of a plane graph G, vertices are realized by internally disjoint elements from a family 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 F-contact representations we insist that at each time step (continuously throughout the morph) we have an F-contact representation. We focus on the case when F is the family of triangles in R2 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 O(n2) 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.
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2023
Angelini, P., Chaplick, S., Cornelsen, S., Da Lozzo, G., Roselli, V. (2023). Morphing Triangle Contact Representations of Triangulations. DISCRETE & COMPUTATIONAL GEOMETRY [10.1007/s00454-022-00475-9].
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Utilizza questo identificativo per citare o creare un link a questo documento: `https://hdl.handle.net/11590/433567`
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