Here the graph product is defined as in the Hatcher.
Anyway I am posting a response by my professor which has answered my question:
In algebraic geometry, blowing up is a process of replacing a point inan algebraic variety by a projective space. In graph products, it is the process of replacing each vertex v of the graph by K(G_v,1), and each edge by the appropriate mapping cylinder (as described in Hatcher).
See section 1B on K(G,1) spaces for further detail.
Adeel
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