  Given a graph of C*-algebras, we prove a long exact sequence in KK-theory for
both the maximal and the vertex-reduced fundamental C*-algebras in the presence
of possibly non GNS-faithful conditional expectations. We deduce from it the
KK-equivalence between the full fundamental C*-algebra and the vertex-reduced
fundamental C*-algebra even for non GNS-faithful conditional expectations. Our
results unify, simplify and generalize all the previous results obtained before
by Cuntz, Pimsner, Germain and Thomsen. It also generalizes the previous
results of the authors on amalgamated free products.

Operator Algebras;K-Theory and Homology