If a graph has two subgraphs that are movable, their union spans the whole graph and their motions coincide on their intersection, we can construct a proper flexible labeling of the whole graph. Same colors of edges mean same lengths.
Graph $S_1$
The motion of the graph $S_{1}$ combines the Dixon I motion of its $K_{3,3}$ subgraph and motion of 3-prism ($L_1$).
Graph $S_2$
The motion of the graph $S_{2}$ combines the Dixon II motion of its $K_{3,3}$ subgraph and motion of $Q_1$. We remark that the triangle 1,2,7 is degenerated.
There is also a collision free linkage that models this motion:
Graph $S_3$
The motion of the graph $S_{3}$ also combines the Dixon II motion of its $K_{3,3}$ subgraph and motion of $Q_1$. The triangle 1,2,7 is again degenerated.
Graph $S_4$
The motion of the graph $S_{4}$ comes easily from a motion of its $K_{3,3}$ subgraph.
