Time Machine and External Drives

[Melodeath]

I used Time Machine to backup my Macbook Pro habitually. I also had it set to backup all of my external hard drives. I have since moved on to an iMac. I connected the "old" Time Machine drive to the new computer, and am using it as this machine's Time Machine drive, as well. I have also now connected all of my previous external HDs to this new computer. If I set TM to include my externals in the backup of this computer, is it going to incrementally back them up from the backups that already exist from when they were connected to the laptop, or is it going to start a brand new backup for each of them since they are connected to a new computer? That would be a big waste of space. Thanks!