  Given a ribbon graph $\Gamma$ with some extra structure, we define, using
constructible sheaves, a dg category $CPM(\Gamma)$ meant to model the Fukaya
category of a Riemann surface in the cell of Teichm\"uller space described by
$\Gamma.$ When $\Gamma$ is appropriately decorated and admits a combinatorial
"torus fibration with section," we construct from $\Gamma$ a one-dimensional
algebraic stack $\widetilde{X}_\Gamma$ with toric components. We prove that our
model is equivalent to $Perf(\widetilde{X}_\Gamma)$, the dg category of perfect
complexes on $\widetilde{X}_\Gamma$.
