For a connected planar simple graph G=(V, E) with e=|E|=16 and v=|V|=9, then find the number of regions that are created when drawing a planar representation of the graph?
Question
For a connected planar simple graph G=(V, E) with e=|E|=16 and v=|V|=9, then find the number of regions that are created when drawing a planar representation of the graph?
Solution
To find the number of regions created when drawing a planar representation of the graph, we can use Euler's formula for planar graphs. Euler's formula states that for a connected planar simple graph with v vertices, e edges, and r regions, the formula is given by:
v - e + r = 2
Given that v = 9 and e = 16, we can substitute these values into the formula:
9 - 16 + r = 2
Simplifying the equation, we have:
-7 + r = 2
Adding 7 to both sides, we get:
r = 9
Therefore, when drawing a planar representation of the graph, there will be 9 regions created.
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