Year: 2026
Type: preprint
Abstract: Phylogenetic networks generalize phylogenetic trees as models of evolutionary history, allowing lineages to merge as well as to diverge. For many types of genetic data the root position of such a network cannot be recovered, so that only a semi-directed network can be inferred: a mixed graph in which only the edges entering a reticulation vertex are directed. A common strategy for inferring such a network is to first infer the subnetwork it induces on each set of $k\geq 3$ of its leaves, called a $k$-net, and then to assemble these pieces. This can only succeed if the $k$-nets determine the network, in which case that network is said to be encoded by its $k$-nets. Semi-directed networks of level-1 and 2, those whose biconnected components contain at most one, respectively two, reticulations, are known to be encoded by their $4$-nets, or quarnets, whereas level-3 networks are not. Even so, in this paper we show that level-3 semi-directed networks are encoded by their $5$-nets, or quinnets, and we characterize the limitation of quarnets exactly: we show that a single previously reported counterexample captures the only obstruction, every other level-3 network being encoded by its quarnets. Our proofs rest on a collection of encoding results for individual structural features of a network, which we establish for networks of arbitrary level and which are of independent interest.
Source: arXiv (Cornell University)
Institutions: Delft University of Technology, University of East Anglia
Language: English
Subfield: Molecular Biology
Domain: Life Sciences
Open Access status: green
OpenAlex ID: W7221037038