  For piecewise-linear maps, the phenomenon that a branch of a one-dimensional
unstable manifold of a periodic solution is completely contained in its stable
manifold is codimension-two. Unlike codimension-one homoclinic corners, such
`subsumed' homoclinic connections can be associated with stable periodic
solutions. The purpose of this paper is to determine the dynamics near a
generic subsumed homoclinic connection in two dimensions. Assuming the
eigenvalues associated with the periodic solution satisfy $0 < |\lambda| < 1 <
\sigma < \frac{1}{|\lambda|}$, in a two-parameter unfolding there exists an
infinite sequence of roughly triangular regions within which the map has a
stable single-round periodic solution. The result applies to both discontinuous
and continuous maps, although these cases admit different characterisations for
the border-collision bifurcations that correspond to boundaries of the regions.
The result is illustrated with a discontinuous map of Mira and the
two-dimensional border-collision normal form.
