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Consider the property of a vertex $v$ of a planar graph $G$ that the circular ordering of its edges is the same (upto orientation) for every graph embedding $\pi$ of $G$ into the plane $\mathbb{R}^2$.

1. Does this property have an official name?

2. (How) can it be defined purely combinatorially?

3. (How) can planar graphs be characterized in which every vertex has this property?

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# Unique circular ordering of edges around a vertex

Consider the property of a vertex $v$ of a planar graph $G$ that the circular ordering of its edges is the same (upto orientation) for every graph embedding $\pi$ of $G$ into the plane $\mathbb{R}^2$.

• Does this property have an official name?

• (How) can it be defined purely combinatorially?

• (How) can planar graphs be characterized in which every vertex has this property?