$\mathbf {The \ Problem \ is}:$ Construct a $2-$sphere with $v$
0-cells, $e$ 1-cells and $f$ 2-cells whenever $v-e+f=2.$
$\mathbf {My \ approach}:$ I am getting stuck in the case of $f \gt 2.$
When, $v=1,e=2,$ the 1st skeleton is $S^1 \vee S^1,$ then also I can't think much (I have attached a diagram what I could try).
Can someone please give a detailed diagram about the case $f \gt 2?$
A small hint is very much needed . Thanks in advance .
$\bullet$ For $v=1$ we have $f=1+e$. So, consider a wedge of $e$-many circles on $\Bbb S^2$ with wedge point as $v$, and then fill up the complement of this wedge by $(1+e)$ many $2$-cells.
$\bullet$ For $v=2$ we have $e=f$. Now, consider two distinct points on $\Bbb S^2$, and join them by $e$-many $1$-cells, i.e. ends of each $1$-cells are the two vertices. Fill the complement by $f=e$ many $2$-cells.
$\bullet$ For $v>2$, take $v$-many points, say $a_1,...,a_v$ on $\Bbb S^2$. Join $a_{i+1}$ with $a_i$ by a $1$-cell for each $i=1,...,v-1$. Now, attach remaining $e-(v-1)$ many $1$-cells such that ends of each of them are $a_1,a_2$. Finally, the complement of whole graph is now have $\big(e-(v-1)\big)+1$ many connected components, fill them by $f=e-v+2$ many $2$-cells.