The complement problem for linklessly embeddable graphs

Abstract

We find all maximal linklessly embeddable graphs of order up to 11, and verify that for every graph [Formula: see text] of order 11 either [Formula: see text] or its complement [Formula: see text] is intrinsically linked. We give an example of a graph [Formula: see text] of order 11 such that both [Formula: see text] and [Formula: see text] are [Formula: see text]-minor free. We provide minimal order examples of maximal linklessly embeddable graphs that are not triangular or not three-connected. We prove a Nordhaus–Gaddum-type conjecture on the Colin de Verdière invariant for graphs on at most 11 vertices. We give a description of the programs used in the search.