Abstract : A phylogenetic network is a rooted acyclic digraph whose leaves are labeled with a set of taxa. The tree containment problem is a fundamental problem arising from model validation in the study of phylogenetic networks. It asks to determine whether or not a given network displays a given phylogenetic tree over the same leaf set. It is known to be NP-complete in general. Whether or not it remains NP-complete for stable networks is an open problem. We make progress towards answering that question by presenting a quadratic time algorithm to solve the tree containment problem for a new class of networks that we call genetically stable networks, which include tree-child networks and comprise a subclass of stable networks.