I am trying to solve a problem.
$G$ is a graph where eccentricity of vertices $x$ and $y$ is $8$, and rest of the vertices have eccentricity $7$. Also $d(x,y) = 8$. I am trying to make eccentricities of $x$ and $y$ also $7$ by adding a vertex (or two vertices). The old graph must be induced in new graph. I proposed a construction:
Let $w$ be any vertex in $G$ such that $d(y,w) = 3$. Add a new vertex $z$ and make it adjacent to $y$ and $w$.
Is this true that every vertex in this new graph will have eccentricity $7$? For a few examples, I got the desired result. Or is there any other way to construct this(by adding one more vertex)?
PS: As asked by @bof, I tried for a smaller value. Like in the following figure, two vertices have eccentricity three and rest have eccentricity two, I tried to make eccentricity of every as two by adding a new vertex. As required the old graph is induced in new graph.