Hi David:

I talked about your remark in the seminar with Raj
and he convinced me that this is the best way to define
the concepts.

A graph G is a customizer of a graph S= (V,E), 
if  G contains V and if S contains an edge (u,v) 
then G contains a path from u to v. 

A graph G is a refinement-customizer of a graph S =(V,E), 
if  G contains V and if S contains an edge (u,v) 
then G contains a V-pure path from u to v. 

A W-pure path from i to j is a path where i and j 
are in W and none of the inner nodes of the path are in W.

What happened with the transitive closure definition:

A graph G is a customizer of a graph S= (V,E), 
if S is a connected subgraph of the transitive closure of G.

We felt that this is more complex to parse.

-- Karl