In an undirected connected planar graph \( G \), there are eight vertices and five faces. The number of edges in \( G \) is \(\underline{\hspace{2cm}}\).
Show Hint
Correct Answer: 11
Solution and Explanation
Step 1: Use Euler's formula for planar graphs.
For a connected planar graph, Euler's formula is given by:
\[
V - E + F = 2
\]
where \( V \) is the number of vertices, \( E \) is the number of edges, and \( F \) is the number of faces.
Step 2: Substitute the given values.
Here, \( V = 8 \) and \( F = 5 \). Substituting into Euler's formula:
\[
8 - E + 5 = 2
\]
Step 3: Solve for \( E \).
\[
13 - E = 2
E = 11
\]
Step 4: Final result.
Thus, the number of edges in the graph is \( 11 \).
% Final Answer
Final Answer: \[ \boxed{11} \]
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