Dual graph of a plane graph has a standard definition https://en.wikipedia.org/wiki/Dual_graph and an edgeless graph on $n$ vertices is planar. What is the standard dual graph of such a graph?
Update from comments:
It seems like $n$ vertex planar edgeless graph could be interpreted as 'some' (I do not know which would be appropriate) 'product' of $1$ vertex graph where the 'product' might have the interpretation that if the underlying graph is connected then the 'product' is connected. Perhaps for such 'product' graphs one might give special meaning to connectedness where usual connectedness fails thus salvaging some definition of duality?