For a graph $G$, $S \subseteq V(G)$ is a dominating set of $G$ if every vertex in $V \backslash S$ is adjacent to at least one vertex in $S$. The connected domination number $\gamma_c(G)$ of $G$ is the minimum cardinality of dominating sets $S$ of $G$ which induce a connected subgraph $G[S]$ of $G$. Non-compliant graphs are graphs such that both the graph and its complement have “large” connected domination numbers. We present some sharp bounds for $\gamma_c(G)$ and use them to tackle questions about intrinsic knotting in graphs and their complements.