A combinatorial condition is obtained for when immersed or embedded incompressible
surfaces in compact 3–manifolds with tori boundary components remain
incompressible after Dehn surgery. A combinatorial characterisation of hierarchies is
described. A new proof is given of the topological rigidity theorem of Hass and Scott
for 3–manifolds containing immersed incompressible surfaces, as found in cubings of
non-positive curvature.