Dyadic representations of graphs

A graph G is dyadic provided it has a representation v → S v from vertices v of G to subtrees S v of a host tree T with maximum degree 3 such that
(i)v and w are adjacent in G if and only if S v and S w share at least three nodes and
(ii) each edge of T is used by exactly two representing subtrees.
We show that a connected graph is dyadic if and only if it can be constructed from edges and cycles by gluing vertices to vertices and edges to edges.