From cycles $C_n$, $n\geq6$, I was trying to form a new graph by adding a single vertex to $C_n$ so that the added vertex has eccentricity two and rest have three. I tried for $C_6$ and $C_7$ as given. But unable to do the same for $C_8$. Kindly suggest me how to proceed. Any hint will be appreciated. Thanks a lot for the help.
P.S. In the figures, red vertex has the eccentricity two.