You cannot "delete the game" from a Nintendogs cartridge. DS games are **mask ROM** — data is physically manufactured onto the chip. It's permanent.
This erases **your dogs, money, items, progress, trainer name** — returns to "Welcome to Nintendogs!" screen.
This is the only way. No menu option. No system settings access (DS saves on cart, not console).
Original Nintendogs (DS): One save, cartridge-bound, reset via 6-button combo. That's it.