rooster X his sister is the best option, since its technically line breeding and not inbreeding, every other option (though I don't know about cousin) is inbreeding and can cause (to name one) inbreeding depression, its kind of complicated but overall brother x sister mating is the best option.
 
This essay on inbreeding coefficient may be useful to you. Remember that chickens have 72 chromosomes, almost double what humans have, and therefore inbreeding chickens does not necessarily have the same harmful effects when done in moderation.

Line breeding with poultry is an established way to breed to the type and colour people want; often one of the best ways to continue the genetics of a great bird is to mate them to their offspring. With that said, each time this happens, it increases the risk of genetic abnormality, therefore should only be done occasionally.

Breeding to cousins involves a lot less risk of inbreeding depression and its' effects, and I myself use this Spiral method to maximise genetic diversity whist breeding to type.
 
All crosses above are acceptable. Line breeding and inbreeding are near one in the same. The difference being with line breeding your trending away from too much "inbreeding" with breeding plan to maintain genetic diversity. Use of aunts, uncles and cousins to maintain diversity.

If you look at a basic breeding chart you'll have a good idea how it's managed.

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In the chart above they bring in another bird (17) to keep up diversity. That one time infusion would instantly take care of any loss of vigor and hatchability due to many generations of inbreeding. The problem with that is introductions of different line of birds can introduce a bevy of problems. It would be advised to get a bird from one of the parent line stock if you feel the need to bring in new blood. In reality if the original parents are not from the same closed flock line they will not need new blood introduction for many generations. There are closed flocks that have been maintained for decades.
 
All crosses above are acceptable. Line breeding and inbreeding are near one in the same. The difference being with line breeding your trending away from too much "inbreeding" with breeding plan to maintain genetic diversity. Use of aunts, uncles and cousins to maintain diversity.

If you look at a basic breeding chart you'll have a good idea how it's managed.

main.php


In the chart above they bring in another bird (17) to keep up diversity. That one time infusion would instantly take care of any loss of vigor and hatchability due to many generations of inbreeding. The problem with that is introductions of different line of birds can introduce a bevy of problems. It would be advised to get a bird from one of the parent line stock if you feel the need to bring in new blood. In reality if the original parents are not from the same closed flock line they will not need new blood introduction for many generations. There are closed flocks that have been maintained for decades.
The chart shows the possible crosses starting from 2 different blood lines. Let's imagine that you have the fortune to find a little flock of a rare breed but they are all brothers and sisters so their genetic pool is identical, what can you do in order to mantain vigor and hatchability without the introduction of new blood?
 
First off the birds are not likely to all be brother and sisters. Half siblings is more apt as it's likely the cock was over multiple hens. Better would be if a person kept several pens and provided eggs from each pen. Ya know, there are ways to get the most out of starting a line and it starts with a good breeder setting you up well.

After hatching and culling down to breeders you'd treat them just like they were separate genetics as no two birds have exactly the same. By line breeding you'd maintain diversity with that small initial gene pool. You can keep going until you see loss of vigor and that will show up in hatchability years down the road. When your hatch rate declines no matter what pairing in your flock you try you know it's time to get in a new cock or hen. No telling when that would be, much depends on initial birds and how well you line bred. If it ever happened you'd be best to locate a breeder cock from the original breeder. Voila! your vigor is restored and hatch rates back.
 
When I started out with genetics (Asiatic Parrots) someone gave me some very simple but easy to understand advice that I have never forgotten. I do accept that is an extremely simplified concept but it is good for a start.

This concept applies to those genetic traits that can be passed from both mom and dad and can be received by both male and female chicks.

The concept is that any trait that is received from both parents so that the baby has two copies of the trait is going to be expressed in a greater way in the baby. Consider it a multiplication effect.

In breeding or Line breeding is a good example of people using the above concept. They are selectively breeding related birds that are known to share a common "desired" trait to increase or improve the expression of that trait in the babies.

The problem is that this concept also applies to undesired traits. If a baby receives a double dose of a negative trait then we end up with a baby that is likely to experience the multiplied effect of that negative trait.

So the success (or failure) of in breeding or line breeding is ultimately going to be dependent upon the presence of both the positive and negative traits in the birds you wish to breed - and this is where your knowledge of your birds is going to help you with your decision making as to which birds to use.

As an additional note: I have always adopted the breeding policy of alternating each generation of breeding between in line and outcrossing. On each alternate generation, I endeavour to outcross to a completely unrelated, strong, well structured bird. This way I am always working towards my desired trait but not compromising the quality of my birds at the end. I have seen too many people do multiple consecutive line-breeds which result in outstanding colours but where their birds are also small, weak and with a whole library of undesirable issues. It is my opinion that it is best to take a few more generations but end up with birds you can be proud of!
 

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