  In the study of rooted phylogenetic networks, analyzing the set of rooted
phylogenetic trees that are embedded in such a network is a recurring task.
From an algorithmic viewpoint, this analysis almost always requires an
exhaustive search of a particular multiset $S$ of rooted phylogenetic trees
that are embedded in a rooted phylogenetic network $\mathcal{N}$. Since the
size of $S$ is exponential in the number of reticulations of $\mathcal{N}$, it
is consequently of interest to keep this number as small as possible but
without loosing any element of $S$. In this paper, we take a first step towards
this goal by introducing the notion of a non-essential arc of $\mathcal{N}$,
which is an arc whose deletion from $\mathcal{N}$ results in a rooted
phylogenetic network $\mathcal{N}'$ such that the sets of rooted phylogenetic
trees that are embedded in $\mathcal{N}$ and $\mathcal{N}'$ are the same. We
investigate the popular class of tree-child networks and characterize which
arcs are non-essential. This characterization is based on a family of directed
graphs. Using this novel characterization, we show that identifying and
deleting all non-essential arcs in a tree-child network takes time that is
cubic in the number of leaves of the network. Moreover, we show that deciding
if a given arc of an arbitrary phylogenetic network is non-essential is
$\Pi_2^P$-complete.
