For a finite undirected graph (V,E) the corresponding graph group G(V,E) is defined as the quotient of the free group generated by the vertex set V modulo the set of all relations of the form ab=ba, where (a,b) is an edge from E.

Graph groups are also known as right-angled Artin groups or a free partially commutative groups. It will be shown that the membership problem in a finitely generated submonoid of a graph group G(V,E) is decidable if and only if (V,E) is a transitive forest. The latter is equivalent to the fact that (V,E) neither contains an induced cycle on four nodes nor an induced simple path on four nodes.

In particular, together with a recent result of Kapovich, Weidmann, and Myasnikov we obtain the first example of a finitely generated group with a decidable generalized word problem that does not have a decidable membership problem for finitely generated submonoids.

We also show that the rational subset membership problem is decidable for the graph group G(V,E) if and only if (V,E) is a transitive forest. This answers a question of Kambites, da Silva, and Steinberg.