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Finding the number of regions in the graph
Problem Statement
Let 'G' be a connected planar graph with 20 vertices and the degree of each vertex is 3. Find the number of regions in the graph.
Solution
By the sum of degrees theorem,
20 ∑ i=1 deg(Vi) = 2|E|
20(3) = 2|E|
|E| = 30
By Euler’s formula,
|V| + |R| = |E| + 2
20+ |R| = 30 + 2
|R| = 12
Hence, the number of regions is 12.
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