  The $sl(2)$ minimal theories are labelled by a Lie algebra pair $(A,G)$ where
$G$ is of $A$-$D$-$E$ type. For these theories on a cylinder we conjecture a
complete set of conformal boundary conditions labelled by the nodes of the
tensor product graph $A\otimes G$. The cylinder partition functions are given
by fusion rules arising from the graph fusion algebra of $A\otimes G$. 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
$(A_4,D_4)$ or 3-state Potts model.
