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The circle packing theorem (Koebe–Andreev–Thurston theorem) claims for a planar graph, we can pack disjoint circles, such that: the circles correspond to vertices and the disks are tangent if the vertices are adjacent.

I would like to implement this algorithm in computer code (input: graph, output: circle packing). Where can I find a readable (not too complex) version of the procedure? I do not need the theory behind it. I know it has been implemented (e.g. in KnotTheory package in Mathematica), but I'm interested in the algorithm itself, not a software that does it.

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Here is Ken Stephenson's book on the subject: "Introduction to Circle Packing: The Theory of Discrete Analytic Functions". The wikipedia page has many more references - https://en.wikipedia.org/wiki/Circle_packing_theorem

Ah, and see the references given at this very closely related post - Koebe–Andreev–Thurston theorem - where can I find a proof?

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