Links in the Complex of Separable Graphs

We determine the homotopy type of the link of a nonempty face in the complex of not 2-connected graphs on n vertices. Corollaries of our main result are that the complex of not 2-connected 3-uniform hypergraphs on n vertices has the homotopy type of a wedge of spheres of dimension n−4 and that the lattice of block-closed graphs on n vertices is Cohen–Macaulay.