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On encodings of phylogenetic networks of bounded level

<28> Many mainstream phylogenetic network reconstruction approaches work by trying to reconstruct a network from relatively simple combinatorial structures. The assumption is that a network should be encoded by the induced collection of such structures. Developing novel mathematical theory, we show that unless a network satisfies a certain requirement this assumption is incorrect, implying that none of the existing algorithms are guaranteed to reconstruct the correct network from an input data set. As a consequence, our result has sparked a whole new direction of research in phylogenetic network reconstruction (see e.g. Huber and Moulton, Algorithmica, DOI 10.1007/s00453-012-9659-x).