A triangulation of a closed surface is $k$-minimal $(k\geq 3)$ if each edge belongs to some essential $k$-cycle and all essential cycles have length at least $k$. It is proved that the class of $k$-minimal triangulations is finite (up to homeomorphism). As a consequence it follows, without referring to the Robertson-Seymour's theory, that there are only finitely many minor-minimal graph embeddings of given representativity. In the topological part, certain separation properties of homotopic simple closed curves are presented.