  It is shown that the knapsack problem, which was introduced by Myasnikov et
al. for arbitrary finitely generated groups, can be solved in NP for graph
groups. This result even holds if the group elements are represented in a
compressed form by SLPs, which generalizes the classical NP-completeness result
of the integer knapsack problem. We also prove general transfer results:
NP-membership of the knapsack problem is passed on to finite extensions,
HNN-extensions over finite associated subgroups, and amalgamated products with
finite identified subgroups.
