Algebraic curves have a discrete analog in finite graphs. Pursuing this analogy,we prove a Torelli theorem for graphs. Namely, we show that two graphs have thesame Albanese torus if and only if the graphs obtained from them by contracting allseparating edges are 2-isomorphic. In particular, the strong Torelli theorem holds for3-connected graphs. Next, using the correspondence between compact tropical curvesand metric graphs, we prove a tropical Torelli theorem giving necessary and sufficientconditions for two tropical curves to have the same principally polarized tropicalJacobian. By contrast, we prove that, in a suitably defined sense, the tropical Torellimap has degree one. Finally, we describe some natural posets associated to a graphand prove that they characterize its Delaunay decomposition.
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