  In this paper we define homological stabilizer codes which encompass codes
such as Kitaev's toric code and the topological color codes. These codes are
defined solely by the graphs they reside on. This feature allows us to use
properties of topological graph theory to determine the graphs which are
suitable as homological stabilizer codes. We then show that all toric codes are
equivalent to homological stabilizer codes on 4-valent graphs. We show that the
topological color codes and toric codes correspond to two distinct classes of
graphs. We define the notion of label set equivalencies and show that under a
small set of constraints the only homological stabilizer codes without local
logical operators are equivalent to Kitaev's toric code or to the topological
color codes.
