An inverse problem on a metric graph with cycle

Consider a quantum graph consisting of a ring with two attached edges, and assume Kirchhoff-Neumann conditions hold at the internal vertices. Associated to this graph is a Schrödinger type operator $L=-\mathrm{\Delta }+q(x)$ with Dirichlet boundary conditions at the two boundary nodes. Let $\{{\omega }_n^2,{\phi }_n(x)\}$ be the eigenvalues and associated normalized eigenfunctions. Let $v_1$ be a boundary vertex, and $v_2$ the adjacent internal vertex. Assume we know the following data: $\{{\omega }_n^2,{\mathrm{\partial }}_x{\phi }_n(v_1),{\mathrm{\partial }}_x{\phi }_n(v_2)\}\mathrm{.}$ Here ${\mathrm{\partial }}_x{\phi }_n(v_2)$ refers to an outward normal derivative at $v_2$ along one of the edges incident to the other internal vertex. From this data we determine the following unknown quantities: the lengths of edges and the potential functions on each edge.