GATE | GATE-CS-2007 | Question 4

Let G be the non-planar graph with the minimum possible number of edges. Then G has
(A) 9 edges and 5 vertices
(B) 9 edges and 6 vertices
(C) 10 edges and 5 vertices
(D) 10 edges and 6 vertices

Answer: (B)

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 K3,3.

That means K5 and K3,3 are minimum non-planar graphs. These graphs have 5 vertices with 10 edges in K5 and 6 vertices with 9 edges in K3,3 graph.
So, graph K5 has minimum vertices and maximum edges than K3,3.

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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