  In the context of two-dimensional rational conformal field theories we
consider topological junctions of topological defect lines with boundary
conditions. We refer to such junctions as open topological defects. For a
relevant boundary operator on a conformal boundary condition we consider a
commutation relation with an open defect obtained by passing the junction point
through the boundary operator. We show that when there is an open defect that
commutes or anti-commutes with the boundary operator there are interesting
implications for the boundary RG flows triggered by this operator. The end
points of the flow must satisfy certain constraints which, in essence, require
the end points to admit junctions with the same open defects. Furthermore, the
open defects in the infrared must generate a subring under fusion that is
isomorphic to the analogous subring of the original boundary condition. We
illustrate these constraints by a number of explicit examples in Virasoro
minimal models.
