I was doing following construction. We know $C_4$ and $C_5$ are $2$-self-centered graphs. When we add a new vertex $x$ and $y$ to $C_4$ and $C_5$, resp, (shown in fig) the new graph contains exactly two vertices with eccentricity three and rest with eccentricity two. I was curious to know if we propose a similar construction where eccentricity of every vertex is three and only one vertex with ecc two, using the same graphs $C_4$ and $C_5$. By adding one or two vertices such that resultant graphs contain $C_4$ and $C_5$ and as induced subgraphs. Any hint will be appreciated. Thanks a lot.
Note : A self-centered graph is a graph where eccentricity of every vertex is the same.