A new airlines company is planning to start operations in a country. The company has identified ten different cities which they plan to connect through their network to start with. The flight duration between any pair of cities will be less than one hour. To start operations, the company has to decide on a daily schedule.
The underlying principle that they are working on is the following:
Any person staying in any of these $10$ cities should be able to make a trip to any other city in the morning and should be able to return by the evening of the same day.
Suppose the $10$ cities are divided into $4$ distinct groups $\text{G1, G2, G3, G4}$ having $3, 3, 2$ and $2$ cities respectively and that $\text{G1}$ consists of cities named $\text{A, B}$ and $\text{C}$. Further, suppose that direct flights are allowed only between two cities satisfying one of the following:
- Both cities are in $\text{G1}$
- Between $\text{A}$ and any city in $\text{G2}$
- Between $\text{B}$ and any city in $\text{G3}$
- Between $\text{C}$ and any city in $\text{G4}$
Then the minimum number of direct flights that satisfies the underlying principle of the airline is ___________