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Planar graph
Hi
I have this graph
Attachment 24559
now i should check if this graph can be planar.
v  number of vertices
e  number of edges
f  number of faces
So it's hold the theorem "If v ≥ 3 then e ≤ 3v − 6 "
v = 9, e = 15
15 ≤ 21
and should hold v  e + f = 2 from here f = 2  v + e = 2  9 + 15 = 8
so f = 8 now my question is how to easily count faces (regions bounded by edges, including the outer, infinitely large region) of the graph?? :D
thanks :)

Re: Planar graph
You need the graph to be embedded in the plane (that is, drawn with no edges crossing) to have faces defined. Also, I think your graph is not planar. The condition you are using works like this "If the graph is planar, then $\displaystyle e \leq 3v  6$" but the latter doesn't prove the first, that is "If $\displaystyle e \leq 3v  6$, then the graph is planar" is WRONG.