The question is this:
In each case, give the values of $r$ (number of regions), $e$ (number of edges), or $v$ (number of vertices), whichever is not given, assuming that the graph is planar. Then either draw a connected, planar graph with the property, if possible, or explain why no such planar graph can exist.
- 8 vertices and 13 edges
Finding $r$ is easy: $$r = e - v +2 = 13 - 8 + 2 = 7$$ so $r = 7$. This is clear to me.
What's not so clear to me is drawing the graph part.
Can I just draw a circle of 8 vertices connected to each other (circuit-wise) and then make random edges connecting two vertices of the circle willy-nilly (as long as they don't cross with other edges) until I reach 13 edges and have 7 regions?
I've seen some people draw a square instead of a circle and I'm a bit confused on that. Is that right? Is there a specific form this graph must take, or is it isomorphic and the form of the graph doesn't really matter as long as you have 8 vertices, 7 regions, and 13 edges that don't overlap with each other?