The complement of a disconnected graph is necessarily connected, but the converse is not true. For instance, $C_5$ is connected and isomorphic to its complement. The following picture shows a graph and its complement which are both connected (and non-isomorphic).
Is there a name for connected graphs whose complements are also connected? In the same vein as coplanar, I tried searching for "co-connected" graphs but didn't find anything. Are there any useful necessary or sufficient conditions for a connected graph to have this property?
Thanks.
Biconnected? Apparently 2-connected graphs are also called biconnected graphs.
Seems a little greedy, I'm sure you could get away with calling these graphs "biconnected" as long as it is clarified.
A "biconnected" graph, then, always has a unique "modular decomposition" (https://en.wikipedia.org/wiki/Modular_decomposition). Nice.
I'm not sure what else could you say, probably loads, but a necessary condition? I don't know, it's known that almost all graphs are biconnected so being disconnected or codisconnected is actually more of a "rare" property.