Diagonal flips of triangulations on closed surfaces preserving specified properties (Q1125957)

Consider a class \(\mathcal P\) of triangulations on a closed surface \(F^2\), closed under vertex splitting. We show that any two triangulations with the same and sufficiently large number of vertices which belong to \(\mathcal P\) can be transformed into each other, up to homeomorphism, by a finite sequence of diagonal flips through \(\mathcal P\). Moreover, if \(\mathcal P\) is closed under homeomorphism, then the condition ``up to homeomorphism'' can be replaced with ``up to isotopy''.