Edge-minimal graphs of exponent 2

A simple undirected graph G has the me2-property if every pair of distinct vertices of G is connected by a path of length 2, but this property does not survive the deletion of an edge. This article considers graphs that can be embedded as induced subgraphs of me2-graphs by the introduction of additional mutually non-adjacent vertices and suitably chosen edges. The main results concern such embeddings of trees.