Given their potential for fault-tolerant operations, topological quantum states are currently the focus of intense activity. Of particular interest are topological quantum error correction codes (QECCs), such as the surface and planar stabilizer codes that are equivalent to the celebrated toric code. While every stabilizer state maps to a graph state under local Clifford operations, the graphs associated with topological stabilizer codes have remained unknown. In this talk, we show that the toric code graph is composed of only two kinds of subgraphs: star graphs (which encode GHZ states) and half graphs. The topological order is identified with the existence of multiple star graphs, which reveals a connection between the toric code and simple repetition codes. We build on these results to obtain necessary and sufficient conditions for a family of graph states to represent topological QECCs. Several examples will be presented.