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It is known that every planar 4-regular 3-connected graph $G$ admits a realization as a system of circles, i.e., it can be drawn on the plane utilizing a set of circles such that the vertices of $G$ correspond to the intersections and touching points of the circles and the edges of $G$ are the arc segments between those points [1].

Let us fix one particular realization of $G$ as a system of circles (realization need not be unique). Suppose the graph $G$ is also vertex-transitive. Can we say anything more about the system of circles in the realization of $G$? Evidently, the points of intesections and touching points (together) will be pairwise symmetric.
Will there be some kind of symmetry between the circles in the system?

One more question, this time not dealing with vertex-transitve graphs; that is, $G$ is a planar 4-regular 3-connected graph. We know how to get a different plane drawing of $G$ by making a bounded face unbouded.
How does this action translate to the system of circles?
(Figure 1 in [1] gives me a feeling that there is some relation though I don't know much about these systems of circles).

[1] Bekos, Michael A.; Raftopoulou, Chrysanthi N., On a conjecture of Lovász on circle-representations of simple 4-regular planar graphs, J. Comput. Geom. 6, No. 1, 1-20 (2015). ZBL1405.05123.

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This doesn't answer the part about symmetry, but the answer to the second part of your question (how do you preserve circularity and the contacts and crossings between circles while changing which face of the arrangement of circles is outermost) is: Möbius transformation or, more or less the same thing, inversion in a circle.

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