Two-chord framings of maximal trees

We found sufficient conditions for a finite connected graph to have a maximal tree with the following property. There are a numbering of edges, and an injective mapping of the set of all edges of the tree to the set of all pairs of different chords ($=$ edges of the graph not contained in the tree) such that for any pair of chords in the image of the mapping the cycles containing one chord from the pair and containing no other chords intersect along an edge in the preimage and, maybe, along other edges of the tree with smaller numbers. The problem of studying graphs possessing this property appeared in the process of studying the (isotopic) classification problem of embeddings of graphs in $3$-space.