  Brane tilings provide the most general framework in string and M-theory for
matching toric Calabi-Yau singularities probed by branes with superconformal
fixed points of quiver gauge theories. The brane tiling data consists of a
bipartite tiling of the torus which encodes both the classical superpotential
and gauge-matter couplings for the quiver gauge theory. We consider the class
of tilings which contain only tiles bounded by exactly four edges and present a
method for generating any tiling within this class by iterating combinations of
certain graph-theoretic moves. In the context of D3-branes in IIB string
theory, we consider the effect of these generating moves within the
corresponding class of supersymmetric quiver gauge theories in four dimensions.
Of particular interest are their effect on the superpotential, the vacuum
moduli space and the conditions necessary for the theory to reach a
superconformal fixed point in the infrared. We discuss the general structure of
physically admissible quadrilateral brane tilings and Seiberg duality in terms
of certain composite moves within this class.
