I am trying to figure out how to extend Euler's formula, n - e + f = 2, to contain a connected component denoted k. I am new to graph theory so I am not sure if the way I got there is correct or if there is a better way to show it.
I started by converting a regular graph into a connected one. I inserted e = e + k - 1 so that each new edge would connect two components in a graph and I got n - e + f = k + 1 which I know is the right answer. Is there a better/easier way to show how Euler's formula can contain a connected component?
Thanks in advance.