We present a graph-based strategy for representing the computational domain for embedded boundary discretizations of conservation-law PDE's. The representation allows recursive generation of coarse-grid geometry representations suitable for multigrid and adaptive mesh refinement calculations. Using this scheme, we implement a simple multigrid V-cycle relaxation algorithm to solve the linear elliptic equations arising from a block-structured adaptive discretization of Poisson's equation over an arbitrary two-dimensional domain. We demonstrate that the resulting solver is robust to a wide range of two-dimensional geometries, and performs as expected for multigrid-based schemes, exhibiting O(N logN) scaling with system size, N.