The authors analyze the rigidity of “torus with hole” graphs, building on classical results like Cauchy’s rigidity theorem for convex polyhedra. To obtain a torus with hole graph, begin with a triangulation on a torus. Then choose any cycle in this graph that bounds a disc on the torus, and delete all interior edges (the edges that lie within that disc).
The main result of this paper is that they characterize minimal rigidity in such graphs, by showing that the following three conditions are equivalent:

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such a graph is minimally \(3\)-rigid (rigid in a generic 3-dimensional embedding, with equality holding in Maxwell’s necessary condition \(|E| \ge 3|V|-6\));

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it is “\((3,6)\)-tight”;

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it can be constructed from \(K_3\) by vertex splitting.

Along the way, they determine the \(17\) possible forms for the boundary graphs, and show that exactly two of these are not contractible.
The paper includes an appendix that contains additional details on some of the proofs.