The Complement Problem for Linklessly Embeddable Graphs

We find all maximal linklessly embeddable graphs of order up to 11, and verify that for every graph $G$ of order 11 either $G$ or its complement $cG$ is intrinsically linked. We give an example of a graph $G$ of order 11 such that both $G$ and $cG$ are $K_6$-minor free. We provide minimal order examples of maximal linklessly embeddable graphs that are not triangular or not 3-connected. We prove a Nordhaus-Gaddum type conjecture on the Colin de Verdi\`ere invariant for graphs on at most 11 vertices. We give a description of the programs used in the search.