The sl(2) minimal theories are classified by a Lie algebra pair where G is of A-D-E type. For these theories on a cylinder we propose a complete set of conformal boundary conditions labelled by the nodes of the tensor product graph . The cylinder partition functions are given by fusion rules arising from the graph fusion algebra of . We further conjecture that, for each conformal boundary condition, an integrable boundary condition exists as a solution of the boundary Yang - Baxter equation for the associated lattice model. The theory is illustrated using the or three-state Potts model.