Abstract : This paper addresses the problem of efficiently turning the connectivity of an initial triangular mesh into a new one, using combinatorial edge flip operations only, excluding vertex insertion and deletion. More precisely, we provide a practical but provably good solution for determining a sequence of edge flips between two surface triangulations sharing the same topological genus and the same number of vertices, with a one-to-one correspondence between the vertices of the two meshes. This is original in that the sequence of edge flips can be determined without using any canonical geometric or topological configuration, classically acting as a pivot between the two triangulations. The paper identifies the main difficulties that must be overcome by this direct algorithm, putting in place the foundation for a generic strategy. An efficient instantiation of this strategy is then presented, with an algorithm that proceeds by successive conformation of the facets, through a region-growing mechanism. The effectiveness of this approach is illustrated by practical benchmarks.