If we confine our attention to species, such as humans, with two sexes and no selfing, a genealogy is simply a set of individuals and the specification of two binary relations, mother of and father of. Of course, each individual has at most one mother and one father specified and the set must observe the restrictions imposed by the temporal aspects. This can be most conveniently specified as a list of individual -father -mother triplets with either or both of the latter two fields allowed to be null. Genealogies lend themselves very naturally to graphical representation and anyone involved in genealogical work or plant and animal genetics will likely have drawn a pedigree at some stage to keep track of relationships. These relationships follow a logic that allows us to represent and quantify correlations between pairs or sets of individuals, or their genes, and thus lets us develop algebras of relationship that we discuss here. We also consider two representations of genealogies as graphs that are of primary importance in genetics, discuss their properties and significance and consider the correspondence between them.
23.1 GENEALOGIES AS GRAPHS
Perhaps the neatest way of representing a genealogy is as a marriage node graph (Figure 23.1). This is a directed graph G = G(V, E), where the set of nodes V is partitioned into three subsets: M the marriages, f the females and m the males. The union of f and m constitutes I, the set of individuals. The edges, E, which are directed, are similarly partitioned into three subsets, fM with each element from a female to a marriage, mM with each element from a male to a marriage and R with each element 781