Abstract
We propose the notion of integrable boundary in the context of discrete quad-graph systems. The equation characterizing the boundary must satisfy a compatibility equation with the one characterizing the bulk that we called the three-dimensional (3D) boundary consistency. In comparison to the usual 3D consistency condition which is linked to a cube, our 3D boundary consistency condition lives on half of a rhombic dodecahedron. We provide a list of integrable boundaries associated to each quad-graph bulk equation of the classification obtained by Adler, Bobenko and Suris. Then, the use of the term 'integrable boundary' is justified by the facts that there are Bäcklund transformations and a zero curvature representation for systems with a boundary satisfying our condition. We discuss the three-leg form of boundary equations and hence obtain associated discrete Toda models with a boundary. Finally, the connection between the 3D boundary consistency and the set-theoretical reflection equation is established.