TITLE: Densest Graphs with Unique Perfect Matching
QUESTION [10 upvotes]: Given a graph $G$ with $n$ vertices, that has a perfect matching $M$, what is the maximal number of edges that $G$ can have without contradicting the uniqueness of $M$? 
Are examples of such extremal graphs known?

REPLY [7 votes]: An easy construction to achieve the bound of $n^2$ edges for $2n$ vertices: embed the $n$ edges $(v_i,w_i)$ of the 1-factor as parallel segments in the plane. To these $n$ edges, add all $\binom n2 $ possible edges $(v_i,w_j)$ with $i<j$ and all $\binom n2 $ edges $(w_i,w_j)$.
For any 1-factor, each $v_i$ must belong to a different edge, from there it is clear that there is a unique one.