Question

Let G be a simple undirected planar graph on 10 vertices with 15 edges. If G is a connected graph, then the number of bounded faces in any embedding of G on the plane is equal to
1. 3
2. 4
3. 5
4. 6

Answer 1

Correct Answer - Option 4 : 6

The correct answer is "option 4".

CONCEPT:

A planar graph is a graph that can be drawn on the plane in such a way that its edges must intersect only at their endpoints.

In a planar graph, the graph is drawn in such a way that no edges must cross each other.

The graph whose edges overlap or cross each other is known as a Non-planar graph.

A graph to be planar must satisfy the following Euler’s formula:

v - e + f = 2

where

v is the number of vertices

e is the number of edges

f is the number of faces 

CALCULATION

v = 10, e = 15

According to Euler’s formula:

v - e + f = 2 

10 – 15 + f = 2

f = 2 – 10 + 15

f = 7

Out of 7, there will be always one unbounded face.

So, the number of faces is 6.

Hence, the correct answer is "option 4".