Let $G$ be a simple unweighted graph. The *distance* between two vertices $u,v$ in $G$ is the length of a shortest path in $G$ between $u$ and $v$. The *diameter* of $G$, denoted $diam(G)$, is the largest distance between two vertices in $G$. For a natural number $k$, The $k^{\mathrm{th}}$ power of $G$, denoted $G^{k}$, is the graph obtained from $G$ by adding edges between every two vertices $u,v$ where the distance between $u,v$ in $G$ is at most $k$.

Let $\mathcal{F}_{k}$ be the family of all graphs that are the $k^{\mathrm{th}}$ powers of graphs of diameter $k+1$. That is,

$\mathcal{F}_{k} = \{G^{k}\mid diam(G)=k+1\}$

$\mathcal{F}_{1}$ is the set of all graphs of diameter $2$, and for all $k\ge2$, $\mathcal{F}_{k}\subseteq\mathcal{F}_{2}$.

Is something more known about the structure of the graphs in $\mathcal{F}_{k}$ for $k\ge2$? In particular, are these containments strict? Do we get more structure the higher the value of $k$?

Thanks for any ideas, pointers, and/or references on this.