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 {mathcal {F}}-contact representation of a plane graph G, vertices are realized by internally disjoint elements from a family {mathcal {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}}-contact representations we insist that at each time step (continuously throughout the morph) we have an {mathcal {F}}-contact representation. We focus on the case when mathcal {F} is the family of triangles in mathbb {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) 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.