We present a study of combinatorial constructions that are related to understanding the structure of bar-joint frameworks.

The primary objects of study in Chapters 2 and 3 of this thesis are connected (k, l)-sparsity matroids. Taking inspiration from [22], where connected (2, 3)-sparsity matroids are considered, we provide a method of constructing graphs with a connected (2, 2)-sparsity matroid. Throughout these chapters we work in as a purely combinatorial a setting as is practical, minimising invocations to theoretic machinery involving frameworks.

In Chapter 4 we show that the aforementioned method of construction is pertinent to characterising globally rigid frameworks in two-dimensional spaces endowed with non-Euclidean norms. This “natural avenue of research” [8, p.181] builds on the characterisation of rigid graphs in such spaces by Dewar. However, when compared to Dewar’s characterisation, we make use of an additional constraint in order to link these combinatorial methods to the structure of frameworks in these spaces. Specifically, we demand that the norms we consider are analytic.

We then turn our attention, in Chapter 5, away from (k, l)-sparsity matroids and towards labelled graphs. More precisely, we provide a method of constructing a family of labelled graphs designed to satisfy sparsity conditions relevant to the rigidity of frameworks realised on not necessarily concentric spheres. A precise connection between this family of graphs this notion of rigidity is not provided. Such a description would extend work characterising rigid frameworks on concentric spheres [32], [33].

To conclude there is a short chapter suggesting ways this research may be built upon.