A rooted phylogenetic network is a finite directed acyclic graph with a unique root vertex of indegree 0 and taxon-labeled leaves of outdegree 0. Unlike a phylogenetic tree, vertices may have indegree greater than 1, allowing the network to represent reticulate evolutionary events such as hybridization or horizontal gene transfer.
Unfolding a phylogenetic network duplicates shared descendant subgraphs and produces a multi-labeled tree. Conversely, folding identifies isomorphic rooted subtrees of a multi-labeled tree. A network is stable if unfolding and then folding recovers an isomorphic network. Moulton and Spillner (2026) prove that the labelable phylogenetic networks are exactly the stable ones.
Let 
be the set of unrestricted rooted binary phylogenetic networks with four reticulations
on 
labeled taxa. Yu and Zhang (2026) classified the 79 possible component graphs into
ten groups and obtained
 |  |  |
(1)
|
 |  |
(2)
|
for 
,
while 
.
