A Branched Andreev‐Thurston Theorem for Circle Packings of the Sphere

ANDREEV-THURSTON THEOREM. Let r be a triangulation of S which is not simplicially equivalent to a tetrahedron and let <£: r—»[0, \n\ be given, where T denotes the set of edges of x. Suppose that (A) if eu e2, e3 e r (1) form a closed loop on S and if 2?=i ^(e^^n, then (eif e2, e3) are the three edges of a two-simplex of x; (B) if ex, e2, e3) eA e r (1) form a closed loop on S and if 2?=i <&{e,) = 2K, then (ex, e2, e2, e4) form the boundary of the union of two adjacent twosimplexes of x.