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K has 2n regions and B = W implies K has n black regions, which imply that; n = V(GB) = V(GW) = r(GB ) (construction).Now, if q = E(GB) then using Euler formula, we have q = n + r 2.
In practice, using the Euler formula the Euler-Poincare characteristic of this reconstruction can be computed from 2 x 2 configuration counts (Serra, 1982; Rosenfeld and Kak, 1982).