Morphing Contact Representations of Graphs
share
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 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. 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(n^2) 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.
READ FULL TEXT