Graph manifolds bounding rational homology balls

The subject jokingly called "3.5-dimensional topology" concerns itself with the interactions between 3-manifolds and 4-manifolds, asking, given a 3-manifold, what 4-manifolds it is the boundary of. One big question in 3.5-dimensional topology is when a rational homology 3-sphere is the boundary of a rational homology 4-ball. (Guess what? Almost never.) We will discuss this question for a particular class of rational homology 3-spheres described by weighted graphs, presenting some results, conjectures, and ways forward.