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## 11.5 Graphs

It is a useful observation that the local game associated with an eyespace depends only on the underlying graph, which as a set consists of the set of vertices, in which two elements are connected by an edge if and only if they are adjacent on the Go board. For example the two eye shapes:

 ```.. .. and .... ```

though distinct in shape have isomorphic graphs, and consequently they are isomorphic as local games. This reduces the number of eyeshapes in the database `patterns/eyes.db'.

A further simplification is obtained through our treatment of half eyes and false eyes. Such patterns are identified by the topological analysis (see section 11.7 Topology of Half Eyes and False Eyes).

A half eye is isomorphic to the pattern `(!.)` . To see this, consider the following two eye shapes:

 ```XOOOOOO X.....O XOOOOOO and: XXOOOOO XOa...O XbOOOOO XXXXXXX ```

These are equivalent eyeshapes, with isomorphic local games {2|1}. The first has shape:

 ```!.... ```

The second eyeshape has a half eye at `a' which is taken when `O' or `X' plays at `b'. This is found by the topological criterion (see section 11.7 Topology of Half Eyes and False Eyes).

The graph of the eye_shape, ostensibly `....' is modified by replacing the left `.' by `!.' during graph matching.

A false eye is isomorphic to the pattern `(!)` . To see this, consider the following eye shape:

 ```XXXOOOOOO X.Oa....O XXXOOOOOO ```

This is equivalent to the two previous eyeshapes, with an isomorphic local game {2|1}.

This eyeshape has a false eye at `a'. This is also found by the topological criterion.

The graph of the eye_shape, ostensibly `.....' is modified by replacing the left `.' by `!'. This is made directly in the eye data, not only during graph matching.

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