Abstract: We consider a family of manifolds $(X_\varepsilon)_\varepsilon$ that shrinks to a metric graph $X_0$ as $\varepsilon \to 0$, i.e., a topological graph where each edge is assigned a length. A simple example is given by the (smoothed) surface of the $\varepsilon$-tubular neighbourhood of $X_0$. Let $Y_0$ be the set of vertices of degree $1$, and $Y_\varepsilon$ the corresponding boundary of $X_\varepsilon$. Using boundary triples based on first order Sobolev spaces, we can define the Dirichlet-to-Neumann map $\Lambda_\varepsilon(z)$ associated to the boundary $Y_\varepsilon$ in $X_\varepsilon$ and a Laplace-type operator $\Delta_\varepsilon \ge 0$ for $\varepsilon \ge 0$. In particular, for suitable $\varphi$ on $Y_\varepsilon$, we let $\Lambda_\varepsilon(z)\varphi$ be the normal'' derivative of the Dirichlet solution $h$ of $(\Delta_\varepsilon -z) h = 0$ with boundary data $\varphi$. The talk is based on a joint work with J.\ Behrndt.