We present a framework to describe tiling of triangulated surfaces, possibly with boundaries. The M-tiles introduced by our framework may be homeomorphic to a disc or not, and capture both the tiles' shape and the geo- metrical and non-trivial topological information of the original mesh. Some tiling algorithms using a cutting scheme are presented using this framework to describe each intermediate state of the cutting process. In particular, we use tiling with a unique tile homeomorphic to a disc to produce an effective computation of the polygonal schema, and to produce a quadrangulation of the original mesh with running time O(gn2 log n), where n is the number of vertices of the mesh, and g the genus. We show that this algorithm produces the minimal number m = 2g - 1 of quadrangles on a boundaryless mesh. Finally, we give some application results and variations on the algorithms.