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.
Best Answer
I want to note that there is one other graph (and only one up to isomorphism) that meets your conditions on $P_8$:
Here, vertices $7$ and $9$ have eccentricity $3$ and the rest have eccentricity $4$.
I have found $76$ such graphs on $P_9$ though many of these are not unique up to isomorphism. Also, some of the graphs are subgraphs of other such graphs. A couple are listed below.
Above, vertices $1$ and $2$ have eccentricity $3$ and the rest have eccentricity $4$.
Above, vertices $7$ and $10$ have eccentricity $3$ and the rest have eccentricity $4$.
I would be happy to provide more information about the graphs if you want more. Just comment.