Dyadic representations of graphs

A graph G is dyadic provided it has a representation v→Sv from vertices v of G to subtrees Sv of a host tree T with maximum degree 3 such that (i)v and w are adjacent in G if and only if Sv and Sw 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.