I will discuss recent work with Zsuzsanna Dancso and Vivek Shende, in which we study the perverse filtration on the cohomology of the compactified Jacobian of a nodal curve, and the weight filtration on the cohomology of a related moduli of microlocal sheaves on the curve. I will describe the behavior of these filtrations under smoothings and normalizations of the curve, and explain how this is an instance of deletion-contraction relations prevalent in graph theory.