(Non)-relative hyperbolicity of graphs of multicurves

Given a compact, connected, orientable surface, we can define many associated graphs whose vertices represent multicurves in the surface. A first example is the curve graph, which has a vertex for every simple closed curve in the surface and an edge joining two vertices if the corresponding curves are disjoint. Given a graph of multicurves satisfying certain natural properties, we can find that it is in some sense non-positively curved. I will present joint work with Jacob Russell on classifying when such graphs are Gromov hyperbolic, relatively hyperbolic and non-relatively hyperbolic, using in particular the example of the separating curve graph.