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It might be simpler than I believed(even trivial?):

ad 2. The circular ordering of the edges (= neighbours) of a vertex $v$ of a planar graph $G$ is unique (upto orientation) when the neighbours of $v$ lie on a cycle that does not contain $v$.

If they happen to lie on more than one cycle their circular ordering doesn't depend on which.
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It might be simpler than I believed (even trivial?):

ad 2. The circular ordering of the edges (= neighbours) of a vertex $v$ of a planar graph $G$ is unique (upto orientation) when the neighbours of $v$ lie on a cycle that does not contain $v$.

If they happen to lie on more than one cycle their circular ordering doesn't depend on which.
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It might be simpler than I believed (even trivial?):

ad 2. The circular ordering of the edges (= neighbours) of a vertex $v$ of a planar graph $G$ is unique (upto orientation) when the neighbours of $v$ lie on a cycle that does not contain $v$.
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