Abstract
Given the results of Chap. 1 that every descendant may be constructed from a complete family of ancestor genes, we now investigate how the properties of the descendant correspond to the properties of those genes. In particular we consider the properties of Hamiltonicity, bipartiteness and planarity. In all three cases we prove that a descendant may only possess the property if all of its ancestor genes do. In the case of bipartiteness and planarity we also establish sufficient conditions. These results allow us to analyse the properties of a graph by considering its ancestor genes, or alternatively, to construct a graph with desired properties by choosing smaller genes with those properties. We follow each section with a discussion of famous results and conjectures relating to the graph properties, and how the results of this chapter relate to them.
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Notes
- 1.
- 2.
In fact, any breeding operations other than type 3 breeding prevents 3-connectedness in all subsequence descendants, since cubic crackers cannot be destroyed by further breeding. Therefore, if a 3-connected descendant is desired, only type 3 breeding operations may be used.
- 3.
Note that the Grinberg graph on 42 vertices [10] is an equally small example of a planar mutant.
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Baniasadi, P., Ejov, V., Filar, J.A., Haythorpe, M. (2016). Inherited Properties of Descendants. In: Genetic Theory for Cubic Graphs. SpringerBriefs in Operations Research. Springer, Cham. https://doi.org/10.1007/978-3-319-19680-0_2
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DOI: https://doi.org/10.1007/978-3-319-19680-0_2
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