Given a graph of C*-algebras as defined in [Adv. Math. 260 (2014), 233–280], we prove a long exact sequence in KK-theory similar to the one obtained by Pimsner in [Invent. Math. 86 (1986), 603–634] for both the maximal and the vertex-reduced fundamental C*-algebras of the graph 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 by Cuntz, Pimsner, Germain, and Thomsen. They also generalize the previous results of the authors on amalgamated free products.