Question

For a connected planar graph with 10 vertices and 15 edges, how many faces does the graph have?

• A. 5
• B. 6
• C. 7
• D. 8

Hint:

When using Euler's formula, students often forget to count the "outer" region as a face. Always remember that for any planar map, the exterior area is always considered one of the faces.

Answer 1

The Correct Option is C

Concept: The relationship between the fundamental components of a planar graph is governed by Euler's Formula. This formula is applicable to any connected graph that can be drawn in a plane without edges crossing.
• Vertices ($V$): The points or nodes in the graph.
• Edges ($E$): The lines connecting the vertices.
• Faces ($F$): The regions bounded by edges, including the single outer, infinitely large region.
• Euler's Formula: For a connected planar graph, $V - E + F = 2$.

Step 1: Extracting given information from the problem.
The problem provides the following parameters for the graph:
• Number of vertices, \( V = 10 \)
• Number of edges, \( E = 15 \)

Step 2: Applying Euler's Formula to find the number of faces.
We substitute the known values into the equation \( V - E + F = 2 \): \[ 10 - 15 + F = 2 \]

Step 3: Solving the algebraic equation.
First, simplify the left side: \[ -5 + F = 2 \] Now, add 5 to both sides to isolate \( F \): \[ F = 2 + 5 \] \[ F = 7 \] The graph has 7 faces in total.