The topology of one-dimensional invariant sets (attractors) is of great interest. R. F. Williams  demonstrated that hyperbolic one-dimensional non-wandering sets can be represented as inverse limits of graphs with bonding maps that satisfy certain strong dynamical properties. These spaces have "homogeneous neighborhoods" in the sense that small open sets are homeomorphic to the product of a Cantor set and an arc. In this paper we examine inverse limits of graphs with more complicated bonding maps. This allows us to understand the topology of a wider class of spaces that includes both hyperbolic and non-hyperbolic attractors. Many of these spaces have the property that most small open sets are homeomorphic to the product of a Cantor set and an arc. The interesting "inhomogeneities" occur away from these neighborhoods. By examining the dynamics of the bonding maps that generate these spaces, we characterize the inhomogeneities, and we show that there is a natural nested hierarchy in the collection of these points that is topological.