Minimal edge extensions of some precomplete graphs

We consider the minimal edge $k$-extensions of precomplete graphs – graphs in which there is a vertex adjacent to all other vertices. The lemma about the marginal value of $k$ when a precomplete graph can have a minimal edge $k$-extension is proved. The full description of all the minimal edge $k$-extensions of precomplete graphs being joins of a complete graph and an empty graph, a chain or a cycle is given.