In low-dimensional topology, a boundary-incompressible surface is a two-dimensional surface within a three-dimensional manifold whose topology cannot be made simpler by a certain type of operation known as boundary compression. Suppose M is a 3-manifold with boundary. Suppose also that S is a compact surface with boundary that is properly embedded in M,meaning that the boundary of S is a subset of the boundary of M and the interior points of S are a subset of the interior points of M.A boundary-compressing disk for S in M is defined to be a disk D in M such that and are arcs in , with , , and ).