We address a classical problem in low dimensional topology: the classification of tamely embedded, finite, connected graphs G in S\sp3 up to ambient isotopy. In the case that the graph G is homeomorphic to S\sp1, our problem reduces to the embedding problem for knots in S\sp3. Our major result is the existence of a unique isotopy class of longitudes of a cycle for an infinite class of graphs. We then define new invariants for this infinite class of graphs. First we define a longitude l\sbc of a cycle c in G. In contrast to the situation of a knot, for a graph it is quite difficult to canonically select an isotopy class of longitudes, since the mapping class group of a many punctured torus is very large. However we prove that longitudes exist for any cycle in any finite graph and are unique in H\sb1(S\sp3−G;\doubz). This definition of a longitude can be considered an extension of the definition of a longitude of a tamely embedded knot in S\sp3. We describe the specific conditions under which l\sbc is unique in Π, the fundamental group of the graph complement, as well as the class of graphs which possess a basis of unique longitudes. Next, in the situation in which l\sbc is unique for a cycle c in G, we define a sequence of invariants μ¯\sbG which detects whether l\sbc lies in Π\sbn, the n\spth term of the lower central series of Π. These invariants can be viewed as extensions of Milnor's μ¯\sbL invariants of a link L. Although μ¯\sbG is not a complete invariant, we provide an example illustrating that μ¯\sbG is more sensitive than Milnor's μ¯\sbL, where L is the subgraph of G consisting of a link.