For a finite simple graph , say is of dimension , and write , if is the smallest integer such that can be represented as a unit-distance graph in . Define to be \emph{dimension-critical} if every proper subgraph of has dimension less than . In this article, we determine exactly which complete multipartite graphs are dimension-critical. It is then shown that for each , there is an arbitrarily large dimension-critical graph with . We close with a few observations and questions that may aid in future work.