Level-$2$ networks from shortest and longest distances

TL;DR

This paper demonstrates that level-2 phylogenetic networks can be reconstructed from their shortest and longest distance matrices, extending previous limitations and identifying conditions for unique reconstruction.

Contribution

It shows that level-2 networks are reconstructible from shortest and longest distance matrices, and characterizes when they are reconstructible from shortest distances alone.

Findings

Level-2 networks can be reconstructed from shortest and longest distance matrices.

Reconstructibility from shortest distances depends on absence of certain subgraphs.

Networks with a leaf on every generator side are reconstructible from shortest distances.

Abstract

Recently it was shown that a certain class of phylogenetic networks, called level-$2$ networks, cannot be reconstructed from their associated distance matrices. In this paper, we show that they can be reconstructed from their induced shortest and longest distance matrices. That is, if two level-$2$ networks induce the same shortest and longest distance matrices, then they must be isomorphic. We further show that level-$2$ networks are reconstructible from their shortest distance matrices if and only if they do not contain a subgraph from a family of graphs. A generator of a network is the graph obtained by deleting all pendant subtrees and suppressing degree-$2$ vertices. We also show that networks with a leaf on every generator side is reconstructible from their induced shortest distance matrix, regardless of level.