We investigate the homotopy type of a variety of families of configurations of graphs in R3 and S 3 . Preliminary results give that the linear configurations of the tetrahedral graph in R3 has the homotopy type of the double mapping cylinder SO(3)/A4 ← SO(3)/A3 [RIGHTWARDS ARROW] SO(3)/S3 , for An the alternating group and Sn the symmetric group. Two presentations and an action on the free group are given. This result is generalized to two families of configuration spaces of codimension 2 and 3 skeleta of simplices in Rn . The final segment is toward understanding the space of unknotted smooth embeddings of spatial planar graphs.