Currently I'm battling with generalizing the concept of dual graphs to infinite graphs. Considering algebraic duals, that is a Graph $G^{\ast}$ for a graph $G$ such that for each cycle in one there is a minimal cut in the other graph, there exists an algebraic dual to a given graph $G$, if and only if two vertices in $G$ are joined by only finitely many edge-disjunct paths. The details can be found in Henning Bruhn's doctoral thesis here.
I just noticed that he a) moved to Tokyo and b) wrote a Habilitation. Good for him, I'll check it out in time.
But back to graphs, the classical notion of a dual graph is a geometric one and is well-enough described in the according Wikipedia article. A famous theorem by Whitney states, that algebraic and geometric dual coincide in the sense, that a graph is planar iff it has an algebraic dual. However, it seems not unlikely that for infinite graphs this does not need to hold.
In another post, I'll try to describe my approach to geometric dual graphs.
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