A non-trivial method can be to calculate the convex hull of the vertices. (This can only be used if the planar graph generated has a convex outer face, which is most likely the case for most of the graph drawing softwares.)

It seems to be an overkill though but it will give you the face exactly in form of a sequence of vertices given the coordinates of the vertices. It takes $O(n\log n)$ time to find it. You can use the standard LEDA library to find it easily using its subroutine.

However it can also be performed in linear time. Please refer to this.

A non-trivial method can be to calculate the convex hull of the vertices. (This can only be used if the planar graph generated has a convex outer face, which is most likely the case for most of the graph drawing softwares.)

It seems to be an overkill though but it will give you the face exactly in form of a sequence of vertices given the coordinates of the vertices. It takes $O(n\log n)$ time to find it. You can use the standard LEDA library to find it easily using its subroutine.

However it can also be performed in linear time. Please refer to this.