It's called inbreeding
if it causes problems, but it's called linebreeding
if it doesn't.
It's all the same thing, and something with which I don't agree personally. I think there need to be outcrossings to keep genetic and mental problems at bay.
You can see the dangers of inbreeding in the Ansata horses to this day, as well as some of the other SE strains. They started with some lovely horses, and wound up with crazy dingbats who had genetic issues because of all the inbreeding, um, linebreeding
family tree should look like a straight stick with no branches. That goes for horses as well as people.