I was trying to construct the graphs $G$ and $H$ by using the cycle graph $C_9$ (or $C_n$) and $H$ and $G$ are induced in $C_n$.
Graph $G$ is formed by attaching a vertex $x$ to $C_9$ (or $C_n\geq 7$) and by making it adjacent to vertex $1$ and all the vertices $j$, where $3\leq j\leq 8$ (or $3\leq j\leq n-1$). Here, in $G$, vertices $1$ and $x$ have eccentricity two and rest of the vertices have eccentricity three, shown in the following figure.
Similarly, I want to draw a graph $H$, where exactly two vertices have eccentricity three and rest of the vertices have eccentricity four.
The graph $H$ must be formed by appending exactly one vertex to $C_n$.
Kindly help. Any hint will be of great help.
P.S. The construction for graph $G$ is for all cycles for $n\geq9$.