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To proceed we declare what will be considered a border from now on. 

Definition. Given a connected graph G = ( P , L ) , we can define a border B as a set of bonds that separates exactly two connected subgraphs. It means that two sets P$_{1}$ , P$_{2}$ ⊂ P exist such that 

• P$_{1}$ ∩ P$_{2}$ =  and P$_{1}$ ∪ P$_{2}$ = P , 

• any path on G from a point of P$_{1}$ to a point of P$_{2}$ must contain at least one bond of B , 

• a path exists between any two points in P$_{i}$ ( i = 1 , 2) that doesn’t contain any bond of B . 

It is noteworthy that the union of two disjoint borders is not a border itself under this definition, as it divides the graph into three subgraphs. This is a feature we’ll later need to avoid overcounting different configurations. 

The intuitive idea of open and closed border is actually an artifact created by our visualiz- ing regular lattices as immersed in a finite dimensional real space: the seeming adjacency of the vertices creates a contour of the graph, which we use to define closed and open borders. The fact is that this contour is heavily dependent on what particular immersion we employ, and ceases to exist when we consider the graph for itself. The border in itself has no geom- etry whatsoever, since it is just a collection of links, and even the notion of ”continuous” border, without further specifications, makes no sense from a graph-theoretic point of view: in a general graph, a border is just a collection of links that splits it into two parts. We now define open and closed borders with respect to an external set of points, as it will be useful later. 

Definition. Given a border B and a set of points E ⊂ P , we say that B is closed with respect to the external points set E if either P$_{1}$ ∩ E =  or P$_{2}$ ∩ E =  , otherwise B is open . 

For any finite subgraph G$_{N}$ of a given graph G , we choose the natural set of external points E : 

<img src='content_image/373102.jpg'>

Now, given a border B$_{i}$ that divides G$_{N}$ into two subgraphs A$_{i}$ and C$_{i}$ , we define A$_{i}$ as 

