let $G(E,V)$ be a graph. I'm trying to find a bijection between the two following sets:
Set A - simple, undirected graphs with $n$ vertices which does not have vertices with a rank of $0$.
Set B - simple, undirected graphs that does not have vertex with a rank of $n−1$.
Both of the sets are finite.
Prove with a bijection that $|A| = |B|$.
If $K(V)$ denotes the complete graph on vertex set $V$, then a suitable bijection is $G(E,V)\mapsto K(V)-G(E,V)$ (works in both directions).