Let p be an integer so that p≥3 and let G be a planar graph having p vertices and 3p−7 edges. Prove that G is connected.

I'm a little unsure of where to begin with this problem. It was suggested that I try to use Euler's Formula which states:

Let G be a connected graph with p verticies and q edges. If G has a planar embedding with f faces then p−q+f=2

I'm not sure how helpful this is since it doesn't prove connectedness, but is just a property of connected graphs with planar embeddings.

Question

Let p be an integer so that p≥3 and let G be a planar graph having p vertices and 3p−7 edges. Prove that G is connected.

I'm a little unsure of where to begin with this problem. It was suggested that I try to use Euler's Formula which states:

Let G be a connected graph with p verticies and q edges. If G has a planar embedding with f faces then p−q+f=2

I'm not sure how helpful this is since it doesn't prove connectedness, but is just a property of connected graphs with planar embeddings.