…or does it remain only connected to those stars it was connected to at the beginning?

A: Yes, it immediately connects. When half of a binary star is destroyed, the star map changes dramatically, and this can really change the situation. You might even want to move what’s left of your homeworld to another spot on the board, to better show the connections, but be sure to mark the star in some way at that point so you can’t forget which one is your Earth.