  We give necessary and sufficient criteria for elementary operations in a
two-dimensional terrain to preserve the persistent homology induced by the
height function. These operations are edge flips and removals of interior
vertices, re-triangulating the link of the removed vertex. This problem is
motivated by topological terrain simplification, which means removing as many
critical vertices of a terrain as possible while maintaining geometric
closeness to the original surface. Existing methods manage to reduce the
maximal possible number of critical vertices, but increase thereby the number
of regular vertices. Our method can be used to post-process a simplified
terrain, drastically reducing its size and preserving its favorable properties.
