I had tried this once before, but had lost all of my notes and screenshots, so I just nuked the vault and killed everyone in it. But I tried again recently, and made sure to save everything this time. But I'm getting ahead of myself.
I saw a post on imgur about somebody who had gotten James as a vault dweller and only used him to mate with the women so everyone in his vault was a child of James, he sent everyone who hadn't slept with, or came from James out into the wasteland to die. I thought it was interesting, now that we've been given the chance to create and run our own vault, how some people were kind of taking the position of Vault-Tec and turning their vaults into social experiments. So I decided that I would create a vault, choose two people that I thought were worthy of breeding, and those two, along with their children were the only ones allowed to have children. But here's the twist. They had to have as much incistuous six as possible. Obviously, in Fallout Shelter you can't make direct siblings (i.e. brothers/sisters, mother/son, father/daughter) have six, but you can make cousins, uncle/niece, aunt/nephew have six. And so that's what I tried to do, only bringing in someone from outside the family when it was absolutely necessary. And so, after just a few days, only building rooms when it was necessary to keep the vault running (making the perfect vault wasn't part of the plan), I spread the family too thin.
It was originally the Day family, Doris and Aaron Day had 4 kids, and Aaron also had a kid with Maria Cox. So I decided, since Doris and Aaron's kids can't have six, I would have one of them create the ability to pro-create incistuously by having kids with Maria. And so, after making everyone have six with their cousins, uncles, and aunts (with the occasional outsider to keep the family going) the Day family was finally gone.
Every time somebody would have six with an outsider, their child's last name would be a mix of the two, and by bringing in too many outsiders (I brought in two, not including Maria from the beginning) the family name was changed to Pratt-Lee with the birth of Walter and Todd Pratt-Lee. Of course I could keep going by just simply making other family members that still keep the name Day breed, but I grew up on Disney movies and am a hopeless romantic, and by doing that than all of the relationships and pseudo-marriages created within this experiment would be broken, and everything would turn to chaos and I truly wish for the Day family to have their happy endings by finding true love.
To keep track of the family, I screenshot every message that popped up saying a baby was born, the one in which you get to change the name of the baby and it shows who the parents are. I also made a family tree using familyecho.com so I could see what the family looked like as a whole instead of just a bunch of screenshots and random people put into random rooms inside the vault.
While choosing the first two people to breed, it would've made sense for me to choose based on their S.P.E.C.I.A.L. but I didn't. I chose based on what they looked like.