  In this work, we answer an open problem in the study of phylogenetic
networks. Phylogenetic trees are rooted binary trees in which all edges are
directed away from the root, whereas phylogenetic networks are rooted acyclic
digraphs. For the purpose of evolutionary model validation, biologists often
want to know whether or not a phylogenetic tree is contained in a phylogenetic
network. The tree containment problem is NP-complete even for very restricted
classes of networks such as tree-sibling phylogenetic networks. We prove that
this problem is solvable in cubic time for stable phylogenetic networks. A
linear time algorithm is also presented for the cluster containment problem.
