Signature bond compatibility

So, given my previous posts on what Faulon's signatures are, here is an explanation of how they are used in the structure enumeration algorithm that I am almost finished implementing.

The core test in this algorithm is for compatible bonds. Two atoms are only joined if : a) they have compatible target signatures and b) there are less than the target number of bonds already. A target signature here is just a signature that is set on the atom for it to match, like a pattern.

The first of these tests is illustrated here:

Another (overly) complex diagram! But the formula here is a bit difficult to interpret otherwise. In the top left corner is a graph G (slightly resembling hexane without the hydrogens) which is, by convention, composed of vertices (V) and edges (E).

The equation to the right of the graph defines part of the condition for a compatible bond. The tau terms are just target signatures, as shown on the upper right. The tricky term is h-1στ(y)(z) which means 'a signature starting from the neighbour z of y in the subgraph defined by τ(y)'. This requires using a target signature (τ(y)) as if it was a subgraph - shown on the bottom left for the target b and the neighbour n.

The same process is repeated on the bottom right of the figure for b and m - which matches the height - 1 target signature for c. This should make sense, since the atoms labelled with b in the graph are attached to both a and c - so the signature for b must be compatible with both. It is easy to check that a and c are not compatible, and cannot therefore be bonded.