To completely remove a frangipani branch from a frangipani tree, you should cut it off around 5mm to 10mm from the fork. Try not to damage the lower feeder branch or the other branch extending from the fork or the fork itself.
If you cut too close to the fork, you risk damaging the fork. If you cut further away from the fork, the remaining branch stump will rot. The rot usually only extends back to the fork so it’s not such a problem, the main problem is the appearance of the rotted knob.
Ideally, the remaining branch stub will harden without rot.