Discussion
If a planar graph has v vertices, e edges and f faces, then which of the following is true ?
Answer: D
The true relation is v - e + f = 2.
This is Euler's formula for a connected planar graph, counting the outer, unbounded face. For example, a triangle has 3 vertices, 3 edges and 2 faces (inside and outside):
\[ 3 - 3 + 2 = 2 \]
Option A happens to give 2 for a single cycle, where v = e, but fails for a single edge: v = 2, e = 1, f = 1 gives e + f - v = 0, while v - e + f = 2. For the triangle, option B gives 3 - 2 + 3 = 4 and option C gives 8.
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