### Abstract

Finding the ancestral recombination graph (ARG) that explains a data set with the least number of recombination is the parsimony-ARG analogue to finding parsimonious phylogenies. This is a hard computational problem and two main approaches will be discussed. Firstly, a "scan along sequences” dynamic programming approach that works up to 10 sequences of any length. Secondly, a "trace history back in time" branch and bound approach that can work very fast for much larger number of sequences, but can also fail totally dependent on data". The second approach can also be extended to include gene conversion. Finally, the number of ancestral states that could be encountered for a given data set it counted for small number of sequences and segregating sites. It is also illustrated how likelihood calculations can be done on a restricted graph that contains close to minimal histories of a set of sequences

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