GATE | GATE-CS-2005 | Question 10

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.

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

â€¢ Number of Faces(F)
â€¢ plus the Number of Vertices (corner points) (V)
â€¢ minus the Number of Edges(E),
always equals 2. This can be written: F + V âˆ’ 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 âˆ’ e + f = 2.

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