Open In App
Related Articles

GATE | GATE-CS-2007 | Question 4

Improve Article
Save Article
Like Article

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


10 edges and 6 vertices


10 edges and 5 vertices 


9 edges and 6 vertices


9 edges and 5 vertices 

Answer: (C)


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.

Quiz of this Question
Please comment below if you find anything wrong in the above post

Level Up Your GATE Prep!
Embark on a transformative journey towards GATE success by choosing Data Science & AI as your second paper choice with our specialized course. If you find yourself lost in the vast landscape of the GATE syllabus, our program is the compass you need.

Last Updated : 04 Feb, 2020
Like Article
Save Article
Similar Reads