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?

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?