Use of Graph Theory in Computation of Inbreeding and Kinship Coefficients

Abstract

The computation of the inbreeding coefficient fronm a given set of individuals whose pedigree is known can be very cumbersome when the pedigree is large and/or complicated. Wright's [1922] original inbreeding coefficient was defined as the correlation between arbitrary values assigned to the uniting gametes. In the present paper we use an alternative definition developed by Haldane and Moshinsky [1939], Cotterman [1941], and Malecot [1948]. Their inbreeding coefficient, f, is defined as the probability that a pair of alleles in the uniting gametes are identical by descent. This f is, however, a special case of Wright's definition, and in fact f = L [2-'(1 + Fa)], where n is the number of individuals in a path connecting the two gametes, Fa is Wright's inbreeding coefficient of the common ancestor in this path, and summation is for all different paths which do not pass through any individual more than once. The kinship coefficient is the probability that two homologous genes taken from two individuals, one from each, are identical by descent (Mal4cot [1948]). Thus the kinship coefficient is the same as the inbreeding coefficient of an offspring if the two individuals are mates. Two different methods have been available to compute the inbreeding coefficient of a particular individual in a given pedigree. One is to count all closed paths in the pedigree which connect the pair or pairs of genes in question, and to measure for each path the probability that the two genes are identical by descent through that path. The inbreeding coefficient is given by adding all the probabilities. The mean inbreeding coefficient for a giveii set of individuals thus can be obtained by taking the average of inbreeding coefficients (Haldane and Moshinsky [1939]). Theoretically, the method can be applied to any type of pedigree, but it is necessary to construct the whole pedigree in advance and to count all paths which contribute to the inbreeding coefficient. Wright and McPhee [1925] have developed a statistical