  Mirror symmetry for higher genus curves is usually formulated and studied in
terms of Landau-Ginzburg models; however the critical locus of the
superpotential is arguably of greater intrinsic relevance to mirror symmetry
than the whole Landau-Ginzburg model. Accordingly, we propose a new approach to
the A-model of the mirror, viewed as a trivalent configuration of rational
curves together with some extra data at the nodal points. In this context, we
introduce a version of Lagrangian Floer theory and the Fukaya category for
trivalent graphs, and show that homological mirror symmetry holds, namely, that
the Fukaya category of a trivalent configuration of rational curves is
equivalent to the derived category of a non-Archimedean generalized Tate curve.
To illustrate the concrete nature of this equivalence, we show how explicit
formulas for theta functions and for the canonical map of the curve arise
naturally under mirror symmetry.
