A graph and its complement with specified properties I: connectivity

We investigate the conditions under which both a graph G and its complement G ˉ possess a specified property. In particular, we characterize all graphs G for which G and G ˉ both (a) have connectivity one, (b) have line-connectivity one, (c) are 2-connected, (d) are forests, (e) are bipartite, (f) are outerplanar and (g) are eulerian. The proofs are elementary but amusing.