Abstract
In Chap. 1 we stated the result that every graph has a unique complete family of ancestor genes. The result is proved in detail in this chapter. The proof is lengthy and is therefore broken up into several intermediate steps. We first show that it is sufficient to prove the result for graphs with no parthenogenic objects. We then consider all possible methods of decomposing a graph into three components via the inverse operations and show that the components so obtained are always the same regardless of the inverse operations used. Next we prove that any complete family of ancestor genes for a graph has cardinality which is a fixed constant for that graph. We then proceed to prove that for any descendant without parthenogenic objects, it is possible to isolate at least two genes with single inverse breeding operations. Finally, we use each of these results to prove the uniqueness theorem.
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© 2016 Springer International Publishing Switzerland
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Baniasadi, P., Ejov, V., Filar, J.A., Haythorpe, M. (2016). Uniqueness of Ancestor Genes. In: Genetic Theory for Cubic Graphs. SpringerBriefs in Operations Research. Springer, Cham. https://doi.org/10.1007/978-3-319-19680-0_3
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DOI: https://doi.org/10.1007/978-3-319-19680-0_3
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-19679-4
Online ISBN: 978-3-319-19680-0
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