Before revision:  Such an injection is called a valid distance drawing. In this document, we study the SCFE problem in real trees (also known as \mathbb{R show that a complete signed graph has a valid distance drawing in a real tree if  and only if  its subgraph composed of all (and only) its positive edges is strongly chordal .
After revision:  Such an injection is called a valid distance drawing show that a complete signed graph has a valid distance drawing in a real tree if and only if its subgraph composed of all (and only) its positive edges is strongly chordal .