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CSE-2005 On the Full-Decomposition Optimality Conjecture for Phylogenetic Networks

Phylogenetic networks are models of evolution that go beyond trees, allowing biological operations that are not consistent with tree-like evolution. One of the most important of these biological operations is (meiotic) recombination between two sequences (homologous chromosomes). The algorithmic problem of reconstructing a history of recombinations, or determining the minimum number of recom-binations needed, has been studied in a number of papers In an earlier paper [?], we established that for any set of sequences M, there is a phylogenetic network deriving M, that has a natural " fully decomposed " structure. That paper also conjectured that there is always a fully decomposed network for M which minimizes the number of recombination used, over all possible phylogenetic networks for M. That conjecture is still open. In this paper, we prove a mathematically weaker, but biologically motivated, version of that conjecture.