I considered a graph, path $P_8$ and added two more vertices such that eccentricity of two vertices is three and rest of the vertices have eccentricity four, and $P_8$ is induced in the new graph. I got the following figure, where exactly two vertices (numbered 2 and 7) are central vertices (eccentricity two) and rest are non-central vertices i.e., diametral vertices (eccentricity four) and $P_8$ is induced in this graph.
Is there any way to get the same result for the graph path $P_9$, or other path graphs, by adding exactly two vertices. Any kind of help or suggestion will be highly useful for me. I am thankful in advance for the help.