Hi Mark,
actually I was trying to find a rule, which does not refer to the
control of the "diagonals" on the board. Instead, I intended to use the
relation of clusters to regions only (without further reference to the
position on the board).
If I understand the "four headed dragon" correctly, there are 4
clusters (say 1 -4, clockwise starting at the top) and 4 regions (say
a-d, clockwise starting at the top-left).
The region a (for example) has two clusters (1 and 4) and therefore
might be a false eye. But 1 and 4 are also linked via 1 - b - 2 - c -
3 - d - 4 and this link does not use region a. Thus, a is a real eye
and so are the other regions.
So I'd say, that my definiton does work in the case of the four-headed
dragon (but indirect (recursive) links between clusters are
necessary). Am I missing something?
As far as I can tell, if you fill in the missing bits of your definition
(as suggested by Robert), it does work.