Inductive constructions for frameworks on a two-dimensional fixed torus

Details Inductive constructions for frameworks on a two-dimensional fixed torus Inductive constructions for frameworks on a two-dimensional fixed torus

2012-03-29

arxiv.org/abs/1203.6561v1

Mathematics

Metric Geometry

35 pages, 13 figures

Scientific paper

In this paper we find necessary and sufficient conditions for the minimal rigidity of graphs on the two-dimensional fixed torus. We use these periodic orbit frameworks (gain graphs) as models of infinite periodic graphs, and the rigidity of the gain graphs on the torus correspond to the generic rigidity of the periodic framework under forced periodicity. Here it is shown that every minimally rigid periodic orbit framework on the two-dimensional fixed torus can be constructed from smaller graphs through a series of inductive constructions. This is a periodic version of Henneberg's theorem about finite graphs. We also describe a characterization of the generic rigidity of a two-dimensional periodic framework through a consideration of the gain assignment on the corresponding periodic orbit framework. This can be viewed as a periodic analogue of Laman's theorem about finite graphs.

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