$\mathcal G_n $ conists of graphs $G$ on $n$ vertices $1,\ldots,n$ such that for all $i<j$, each vertex $k, i<k<j$ is not adjacent to at least one of $i$ and $j$.
Question 1. Can we classify $\mathcal G_n$?
Question 2. Is there any nice family of graph which is a subclass of $\mathcal G_n$? (Seems like path $P_n$, star $S_n$ and probably trees on $n$ vertices are in $\mathcal G_n$) I will appreciate any comments for at least question 2.