The K\H{o}nig Graph Process

Say that a graph G has property $\mathcal{K}$ if the size of its maximum matching is equal to the order of a minimal vertex cover. We study the following process. Set $N:= \binom{n}{2}$ and let $e_1, e_2, \dots e_{N}$ be a uniformly random ordering of the edges of $K_n$, with $n$ an even integer. Let $G_0$ be the empty graph on $n$ vertices. For $m \geq 0$, $G_{m+1}$ is obtained from $G_m$ by adding the edge $e_{m+1}$ exactly if $G_m \cup \{ e_{m+1}\}$ has property $\mathcal{K}$. We analyse the behaviour of this process, focusing mainly on two questions: What can be said about the structure of $G_N$ and for which $m$ will $G_m$ contain a perfect matching?