In this note, we establish necessary and sufficient conditions to determine whether a given embedding of a graph in S3 is ambient isotopic to a planar embedding. Our methods are geometric and our results apply to any tame embedding of any finite, abstractly planar graph. We also consider the problem of whether an associated link determines an embedded graph. For graphs of rank less than three, the associated link is trivial if and only if the graph is planar. For graphs of rank at least three, we prove the existence of nonplanar graphs with trivial associated links using an example of P. Zhao [Topology Appl. 57 (1994), 23-30].