In planar graph, for any vertex v, it's possible to find an embedding in which v is in unbounded face

Question

As mentioned in title, I am wondering..

Assume G is a planar graph and pick any vertex v.

1. Is is possible to find a planar embedding in which v is in an unbounded face?
2. What if we add a constraint that v is the vertex which is connected to all other vertices?

Thank you in advance.

I encountered this problem when I solved a question in Introduction to Graph Theory by West.

Reference - Solution to a problem in Intro to Graph Theory