Moreover, a phylogenetic network that satisfies the third constraint (but not necessarily the first two constraints) is referred to as a [*tree-child*]{} network in the literature [@cardona12]. Intuitively, if a phylogenetic network ${{\mathcal N}}$ is temporal, then one can assign a time stamp to each of its vertices such that the following holds for each edge $(u,v)$ in ${{\mathcal N}}$. If $v$ is a reticulation, then the time stamp assigned to $u$ is the same as the time stamp assigned to $v$. Otherwise, the time stamp assigned to $v$ is strictly greater than that assigned to $u$. Baroni et al. [@baroni06] showed that it can be checked in polynomial time whether or not a given phylogenetic network satisfies the first two constraints. Humphries et al. [@humphries13] have established a new characterization to compute the minimum number of reticulation events that is needed to simultaneously embed an arbitrarily large collection ${{\mathcal P}}$ of rooted binary phylogenetic trees into a temporal network. This characterization, which is formally defined in Section \[sec:prelim\], is in terms of [*cherries*]{}, and the existence of a particular type of sequence on the leaves of the trees, called a [*cherry-picking sequence*]{}. It was shown that such a sequence 