Quadratic differentials and weighted graphs on compact surfaces

We prove that for every simply connected graph $\Gamma$ embedded in a compact surface ${\mathcal{R}}$ of genus $g\ge 0$, whose edges $e_{kj}^i$ carry positive weights $w_{kj}^i$, there exist a complex structure on ${\mathcal{R}}$ and a Jenkins-Strebel quadratic differential $Q(z)\,dz^2$, whose critical graph $\Phi_Q$ complemented, if necessary, by second degree vertices on its edges, is homeomorphic to $\Gamma$ on ${\mathcal{R}}$ and carries the same set of weights. In other words, {\it every positive simply connected graph on ${\mathcal{R}}$ can be analytically embedded in} ${\mathcal{R}}$. As a consequence, we establish the existence of systems of disjoint simply connected domains on ${\mathcal{R}}$ with a prescribed combinatorics of their boundaries, which carry proportional harmonic measures on their boundary arcs.