  We describe a rich family of binary variables statistical mechanics models on
a given planar graph which are equivalent to Gaussian Grassmann Graphical
models (free fermions) defined on the same graph. Calculation of the partition
function (weighted counting) for such a model is easy (of polynomial
complexity) as reducible to evaluation of a Pfaffian of a matrix of size equal
to twice the number of edges in the graph. In particular, this approach touches
upon Holographic Algorithms of Valiant and utilizes the Gauge Transformations
discussed in our previous works.
