Kevin Hartshorn proved that incompressible surfaces in a
closed 3-manifold M provide a limit on distance for a Heegaard
splitting of M. We show there is a relative version that extends to
3-manifolds with boundary. But the main and broader result is this: in
the relative version, bicompressible but weakly incompressible
surfaces can mostly be used instead of essential surfaces. The same is
true in the setting of Heegaard splittings, i. e. the genus of one
Heegaard splitting puts a bound on the distance of any other (ie
non-isotopic) splitting.