Suppose there is a connected planar simple graph G with v vertices such that all its regions are triangles (a cycle consisting of three edges).
How many regions would this planar graph, G, split the plane into?
The graph we are looking at probably looks like a triangular structure that gets more complex as the number of vertices and edges increase, something like this:
I assume the answer will invovle this equation:
r = e - v + 2
Since I am asked to derive the number of regions (r) with only just the number of vertices (v), I must first use v to find the number of edges (e) in order to finally calculate r.
The problem is I don't know how to use v to find e.
I'm sitting on this equation:
e <= 3v - 6
It kind of suggests a connection between e and v, but the relationship is not certain. Am I suppose to use combinotirial math to find the answer?
Any feedback would be appreciated, thanks!
Let $F$ denote the number of faces and $E$ denote the number of edges, then $F=(n-2)*2$.
Or $E=(n-2)*3$ and since $F*3=\sum_{f \in face(G)}{len(f)=2*E}=2(n-2)*3 \Rightarrow F=2*(n-2)$.