If a graph is planar satisfy these constraints,those are
i)r=e-v+2 // from this find 'r'
ii)kr<=2e // substitute here.
iii)e<=k(v-2)/k-2. //'k' is min degree of region
Consider graph G1 assume it is planar
k=4,e=9,v=6.
i)r=9-6+2=5
ii) 4r<=2e
4*5<=2*9 // condition fails so graph is nonplanar.
Consider graph G2 assume it is planar.
k=3,e=11,v=6
i)r=11-6+2=7
ii)3r<=2e
3*7<=2*11 // it satisfying
iii)e<=k*(v-2)/k-2
11<=3*(6-2)
so it satisfying all conditions our assumption is correct.
Hence G1 is nonplanar and G2 is planar.
