If the plane is divided into $\large\color{maroon}{\text{r}}$ regions then, what is the retation between $\large\color{maroon}{\text{k}}$ , $\large\color{maroon}{\text{v}}$ , $\large\color{maroon}{\text{e}}$ and $\large\color{maroon}{\text{r}}$ ?

$\begin{align*} &\text{r}_i = \text{No of finite closed region for component} \; i \\ &\text{Since each component } i \text{ is planar and connected} \\ &\Rightarrow r_i = e_i - v_i + 1 \;\;\; \left \{ \text{Euler formula for only closed region} \right \}\\ &\Rightarrow r = \sum r_i = \sum e_i - \sum v_i + \sum 1 \\ &\Rightarrow r = e - v + k \\ &\Rightarrow r = e - v + k + 1 \;\;\; \left \{ \text{considering outer region also} \right \} \\ \end{align*}$

$$\begin{align*} &\text{r}_i = \text{No of finite closed region for component} \; i \\ &\text{Since each component } i \text{ is planar and connected} \\ &\Rightarrow r_i = e_i - v_i + 1 \;\;\; \left \{ \text{Euler formula for only closed region} \right \}\\ &\Rightarrow r = \sum r_i = \sum e_i - \sum v_i + \sum 1 \\ &\Rightarrow r = e - v + k \\ &\Rightarrow r = e - v + k + 1 \;\;\; \left \{ \text{considering outer region also} \right \} \\ \end{align*}$$