  Phylogenetic networks are becoming of increasing interest to evolutionary
biologists due to their ability to capture complex non-treelike evolutionary
processes. From a combinatorial point of view, such networks are certain types
of rooted directed acyclic graphs whose leaves are labelled by, for example,
species. A number of mathematically interesting classes of phylogenetic
networks are known. These include the biologically relevant class of stable
phylogenetic networks whose members are defined via certain "fold-up" and
"un-fold" operations that link them with concepts arising within the theory of,
for example, graph fibrations. Despite this exciting link, the structural
complexity of stable phylogenetic networks is still relatively poorly
understood. Employing the popular tree-based, reticulation-visible, and
tree-child properties which allow one to gauge this complexity in one way or
another, we provide novel characterizations for when a stable phylogenetic
network satisfies either one of these three properties.
