  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.
