How to Morph Graphs on the Torus

Spaces of Geodesic Triangulations of Surfaces

“…The space of geodesic triangulations of the 2-sphere X (S 2 , T ) was studied by Awartani and Henderson [3], but its homotopy type remains open. The path-connectedness of spaces of geodesic triangulations of flat tori was proved by Chambers et al [9]. If S is a hyperbolic surface, Hass and Scott [18] showed that X (S, T ) is contractible if T is a one-vertex triangulation.…”

On Morphing 1-Planar Drawings

“…We conclude this section with a remark. As already mentioned, Chambers et al [17] studied morphs of toroidal graphs and asked to generalize their result to surfaces of higher genus. We note that, since an n-vertex graph embeddable on a surface of genus g has at most 3n + 6(g − 1) edges, while n-vertex optimal 1-planar straight-line drawable graphs have 4n−9 edges, it follows that the latter do not admit an embedding (without edge crossings) on any surface of bounded genus.…”

“…One way of simplifying the problem is to consider graphs that are non-planar but still admit embeddings on surfaces of bounded genus. In this direction, Chambers et al [17] proved the existence of morphs for pairs of crossing-free drawings on the Euclidean flat torus (where edges are still geodesics). Their technique is rather complex and the authors concluded that an extension to higher genus surfaces is fairly non-trivial.…”

The Deformation Space of Geodesic Triangulations and Generalized Tutte's Embedding Theorem

“…The connectivity of these spaces has been explored in [5,6,14]. Awartani-Henderson [1] identified a contractible subspace in the space of geodesic triangulations of the 2-sphere.…”