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GATE | GATE-CS-2005 | Question 10

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Let G be a simple connected planar graph with 13 vertices and 19 edges. Then, the number of faces in the planar embedding of the graph is

(A)

6

(B)

8

(C)

9

(D)

13


Answer: (B)

Explanation:

An undirected graph is called a planar graph if it can be drawn on a paper without having two edges cross and such a drawing is called Planar Embedding. We say that a graph can be embedded in the plane, if it planar. A planar graph divides the plane into regions (bounded by the edges), called faces. Graph K4 is planar graph, because it has a planar embedding as shown in figure below. 

\"P_graph\" 

Euler\’s Formula : For any polyhedron that doesn\’t intersect itself (Connected Planar Graph),the 

• Number of Faces(F) 
• plus the Number of Vertices (corner points) (V) 
• minus the Number of Edges(E), 
always equals 2. This can be written: F + V − E = 2. 

Solution: 
Here as given, F=?,V=13 and E=19 
-> F+13-19=2 
-> F=8 
So Answer is (B). 

This solution is contributed by Nirmal Bharadwaj 

We can apply Euler\’s Formula of planar graphs. The formula is v − e + f = 2.


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Last Updated : 14 Feb, 2018
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