MLLS-reconstructibility of binary valid networks

A binary valid network is a binary phylogenetic network satisfying the validity conditions considered in the paper. A maximum lower-level subnetwork (MLLS) is obtained by deleting one reticulation edge from each reticulation cycle so that the resulting network is a maximum subnetwork. A network is MLLS-reconstructible if it is uniquely determined by its collection of MLLSs.

MLLS-reconstructibility conjecture. The class of binary valid networks is MLLS-reconstructible.

This conjecture concerns whether binary valid networks can be reconstructed uniquely from their reticulate-edge-deleted maximum lower-level subnetworks. The preceding discussion identifies the reconstruction of a $2$-reticulated cherry as a difficulty because the deleted reticulation edge can have two possible insertion locations; the conjecture asserts that the full collection of MLLSs nevertheless determines the original network.