GATE | GATE-CS-2007 | Question 4

Let G be the non-planar graph with the minimum possible number of edges. Then G has

(A)

10 edges and 6 vertices

(B)

10 edges and 5 vertices 

(C)

9 edges and 6 vertices

(D)

9 edges and 5 vertices 

Answer: (C)

Explanation:

According to Kuratowski\’s Theorem, a graph is planar if and only if it does not contain any subdivisions of the graphs K₅ or K_3,3. 

That means K₅ and K_3,3 are minimum non-planar graphs. These graphs have 5 vertices with 10 edges in K₅ and 6 vertices with 9 edges in K_3,3 graph. 
So, graph K₅ has minimum vertices and maximum edges than K_3,3. 

Alternative method:
A plane graph having ‘n’ vertices, cannot have more than ‘2*n-4’ number of edges. Hence using the logic we can derive that for 6 vertices, 8 edges is required to make it a plane graph. So adding one edge to the graph will make it a non planar graph. 

So, 6 vertices and 9 edges is the correct answer. 

So, option (B) is correct.

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