  We provide a simple, combinatorial criteria for a hierarchically hyperbolic
space to be relatively hyperbolic by proving a new formulation of relative
hyperbolicity in terms of hierarchy structures. In the case of clean
hierarchically hyperbolic groups, this criteria characterizes relative
hyperbolicity. We apply our criteria to graphs associated to surfaces and prove
that the separating curve graph of a surface is relatively hyperbolic when the
surface has zero or two punctures. We also recover a celebrated theorem of
Brock and Masur on the relative hyperbolicity of the Weil-Petersson metric on
Teichmuller space for surfaces with complexity three.
