A structural characterization of tree-based phylogenetic networks
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Attempting to recognize a tree inside a network is a fundamental undertaking in evolutionary analysis. Therefore, the notion of tree-based phylogenetic networks, which was introduced by Francis and Steel, has attracted much attention of researchers in the area of theoretical biology in the last few years. Tree-based networks can be viewed as a natural generalization of rooted binary phylogenetic trees because they are merely trees with additional arcs, and in defining those networks, a certain kind of spanning trees called subdivision trees plays an essential role. In this paper, we provide a structural characterization of tree-based networks that furnishes efficient algorithms for solving the following problems in linear time (for enumeration, in linear delay): given a rooted binary phylogenetic network N, 1) determine whether or not N is tree-based and find a subdivision tree if there exists any (decision/search problem); 2) compute the number of subdivision trees of N (counting problem); 3) list all subdivision trees of N (enumeration problem); and 4) find a subdivision tree to maximize or minimize a prescribed objective function (optimization problem). Our structural result settles numerous questions including the complexity of the problem of counting subdivision trees that was left open in the paper of Francis and Steel, and also provides short proofs of different known results from a unifying point of view. The results in this paper still hold for a certain class of non-binary networks. Some applications and further research directions are also mentioned.
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