A circle contact structure (CCS) is a realization of a graph with vertices representing circles on the plane with edges representing (internal or external unspecified) tangential contacts. In addition, for a natural algebraic description, each circle must be assigned a sign so that

internal contacts occur between circles with opposite sign and external contacts occur between circles with the same sign, in particular excluding more than 2 circles contacting at a point. Such structures have been studied by Bowers, Pratt and Stephenson in the context of inversive distance frameworks.

For a generic circle contact structure (GCCS), there is a neighborhood of radius vectors in which each radius can be independently increased or decreased while preserving the contacts and remaining radii.

We show that graphs with generic circle contact structures are independent in the 2-dimensional generic rigidity (Laman) matroid, extending a result of Connelly Gortler and Theran, who considered (generic) disk contact structures ((G)DCS) i.e. with disjoint interiors, as in the famous Koebe Andreev Thurston theorem.

We answer a further question posed by them: we show that the (possibly nongeneric) bar-joint framework arising from a CCS has no self-stress (its bar-joint rigidity matrix has independent rows) if and only if the CCS is generic.

In the converse direction, there are many graphs, e.g. $K_{3,3}$, that are independent in the Laman matroid, that do not have GCCS. and are also dependent in, e.g., in the hyperconnectivity matroid of Kalai in 2 dimensions, which has both $K_4$ and $K_{3,3}$ as circuits (this matroid has recently been shown by Ruiz and Santos to have several equivalent characterizations as arising from rank 2 skew-symmetric matrix completions of Bernstein, co-factor or spline space matroids of Billera and Whiteley with points on a parabola, and the Laman matroid with points on a moment curve, and conjectured by Jackson and Tanigawa to be the unique free-est matroid - maximal matroid in the weak order poset - with $K_4$ and $K_{3,3}$ as circuits).

We show the set of the graphs underlying GCCS are not the set of independent sets of any matroid over the edges of the complete graph $K_n$. In the process, we show that GCCS graphs are not necessarily independent in the abovementioned equivalent matroids.

In a more positive vein, we show that a subclass of minimally rigid (Laman) graphs that can be inductively constructed by 0-extension or a restricted form of 1-extension (also a restricted form of vertex splitting) have generic radius circle contact structures. However, there are planar Laman graphs that cannot be generated using these operations thereby only partially answering the open question posed in \cite{connelly2019} asking whether all planar Laman graphs have GDCS.

In the process of proving these results, we establish basic properties and tools of CCS, e.g. preserved by moebius transformations, along with extensions of most of our theorems to generic mixed bar and circle contact structures (GBCCS). We additionally show a characterization of graphs $G$ realizable as a GCCS: $G$ has a GCCS if and only if the dimension of the edge length vectors of CCS realizations of $G$ is equal to the rank of $G$ in the even cycle matroid. In particular, if $G$ is generated by the vertex splitting inductive construction (which includes all planar Laman graphs), its rank in the even cycle matroid is $|V(G)|$.

Joint work with Fei Wang.