Display sets of normal and tree-child networks

Phylogenetic trees canonically arise as embeddings of phylogenetic networks. We recently showed that the problem of deciding if two phylogenetic networks embed the same sets of phylogenetic trees is computationally hard, blue{in particular, we showed it to be $Pi^P_2$-complete}. In this paper, we establish a polynomial-time algorithm for this decision problem if the initial two networks consists of a normal network and a tree-child network. The running time of the algorithm is quadratic in the size of the leaf sets.