Morphing Contact Representations of Graphs · arXivDesk
1903.07595Mar 18, 2019Extended version of "Morphing Contact Representations of Graphs", to appear in Proceedings of the 35th International Symposium on Computational Geometry (SoCG 2019)
Morphing Contact Representations of Graphs
Patrizio Angelini, Steven Chaplick, Sabine Cornelsen, Giordano Da Lozzo, Vincenzo Roselli
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
Nearby in the stack
-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. 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 an
n
-vertex plane triangulation, and, if so, computes a morph with
O(n2)
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.