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Choose the correct alternatives ( more than one may be correct) and write the corresponding letters only:

(viii) A non-planar graph with minimum number of vertices has

(a) 9 edges, 6 vertices

(b) 6 edges, 4 vertices

(c) 10 edges, 5 vertices

(d) 9 edges, 5 vertices

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k5, k3,3 which are non planner , but k5 with minimum vertex ... 5, so no of edges n(n-1)/2 = 10 edges .

A non-planar graph with minimum number of vertices has 10 edges, 5 vertices i.e K5

A non-planar graph with minimum number of edges has 9 edges, 6 vertices i.e K3,3

Using  Planarity criteria relation  $e \leq 3\times v -6,$

All other option satisfies this relation except option $(C)$

i.e$10 \nleqslant 3 \times 5-6$

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sir , I think it is not sufficient condition..because  for K3,3 , 9<= 3*6 - 6 but it is non-planar graph..please correct me if I m wrong..