Constructing an Evolutionary Tree and Path–Cycle Graph Evolution along It

The paper solves the problem of constructing an evolutionary tree and the evolution of structures along it. This problem has long been posed and extensively researched; it is formulated and discussed below. As a result, we construct an exact cubic-time algorithm which outputs a tree with the minimum cost of embedding into it and of embedding it into a given network (Theorem 1). We construct an algorithm that outputs a minimum embedding of a tree into a network, taking into account incomplete linear sorting; the algorithm depends linearly on the number of nodes in the network and is exact if the sorting cost is not less than the sum of the duplication cost and the loss cost (Theorem 3). We construct an exact approximately quadratic-time algorithm which, for arbitrary costs of SCJ operations, solves the problem of reconstruction of given structures on any two-star tree (Theorem 4). We construct an exact algorithm which reduced the problem of DCJ reconstruction of given structures on any star to a logarithmic-length sequence of SAT problems, each of them being of approximately quadratic size (Theorem 5). The theorems have rigorous and complete proofs of correctness and complexity of the algorithms, and are accompanied by numerical examples and numerous explanatory illustrations, including flowcharts.