network [@baroni05; @semple07] and, hence, the optimization problem is well-defined. Moreover, despite the problem being NP-hard [@bordewich07], even for when $|{{\mathcal P}}|=2$, several exact algorithms have been developed that, given two rooted phylogenetic trees, construct a phylogenetic network whose number of reticulation events is minimized over the space of all networks that embed both trees [@albrecht12; @chen13; @piovesan12; @wu10].

Motivated by the introduction of temporal networks [@baroni06; @moret04], which are phylogenetic networks that satisfy several time constraints, Humphries et al. [@humphries13; @humphries13a] recently investigated the special case of the aforementioned optimization problem for when one is interested in minimizing the number of reticulation events over the smaller space of all temporal networks that embed a given collection of rooted binary phylogenetic trees. More precisely, in the context of their two papers, the authors considered [*temporal networks*]{} to be phylogenetic networks that satisfy the following three constraints:

(1) speciation events occur successively,

(2) reticulation events occur instantaneously, and

(3) each non-leaf vertex has a child whose in-degree is one.

The second constraint implies that the three species that are involved in a reticulation event, i.e. the new species resulting from this event and its two distinct parents, must coexist in time. 