S. Allen Broughton
A tiling by triangles of an orientable surfaces is called kaleidoscopic if the local reflection in any edge of a triangle extends to a global isometry of the surface. Given such a global reflection the fixed point subset of the reflection consists of embedded circles (ovals) whose union is called the mirror of the reflection. The reflection is called separating if removal of the mirror disconnects the surface into two components. We consider surfaces such that the orientation preserving subgroup of the tiling group generated by the reflection is cyclic or abelian. A complete classification of those surfaces with separating reflection is obtained in the cyclic case as well as partial results for abelian, non-cyclic groups.
Baeth, Nicholas; Deblois, Jason; and Powell, Lisa, "Separability of Tilings" (2001). Mathematical Sciences Technical Reports (MSTR). 105.