Transit functions provide a unifying approach to many results on intervals, convexities, and betweenness. Here we show that hierarchical structures arising in cluster analysis and phylogenetics have a natural characterization in terms of transit functions and that hierarchies are identified by multiple combinations of independent axioms.
k-point crossover operators and their recombination sets are studied from different perspectives. We show that transit functions of k-point crossover generate, for all k > 1, the same convexity as the interval function of the underlying graph. This settles in the negative an open problem by Mulder about whether the geodesic convexity of a connected graph G is uniquely determined by its interval function I. The conjecture of Gitchoff and Wagner that for each transit set R k (x, y) distinct from a hypercube there is a unique pair of parents from which it is generated is settled affirmatively. Along the way we characterize transit functions whose underlying graphs are Hamming graphs, and those with underlying partial cube graphs. For general values of k it is shown that the transit sets of k-point crossover operators are the subsets with maximal Vapnik-Chervonenkis dimension. Moreover, the transit sets of k-point crossover on binary strings form topes of uniform oriented matroid of VC-dimension k + 1. The Topological Representation Theorem for oriented matroids therefore implies that k-point crossover operators can be represented by pseudosphere arrangements. This provides the tools necessary to study the special case k = 2 in detail.
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