  Myasnikov et al. have introduced the knapsack problem for arbitrary finitely
generated groups. In previous work, the authors proved that for each graph
group, the knapsack problem can be solved in $\mathsf{NP}$. Here, we determine
the exact complexity of the problem for every graph group. While the problem is
$\mathsf{TC}^0$-complete for complete graphs, it is $\mathsf{LogCFL}$-complete
for each (non-complete) transitive forest. For every remaining graph, the
problem is $\mathsf{NP}$-complete.
