  A graph is almost self-centered (ASC) if all but two of its vertices are
central. An almost self-centered graph with radius $r$ is called an $r$-ASC
graph. The $r$-ASC index $\theta_r(G)$ of a graph $G$ is the minimum number of
vertices needed to be added to $G$ such that an $r$-ASC graph is obtained that
contains $G$ as an induced subgraph. It is proved that $\theta_r(G)\le 2r$
holds for any graph $G$ and any $r\ge 2$ which improves the earlier known bound
$\theta_r(G)\le 2r+1$. It is further proved that $\theta_r(G)\le 2r-1$ holds if
$r\geq 3$ and $G$ is of order at least $2$. The $3$-ASC index of complete
graphs is determined. It is proved that $\theta_3(G)\in \{3,4\}$ if $G$ has
diameter $2$ and for several classes of graphs of diameter $2$ the exact value
of the $3$-ASC index is obtained. For instance, if a graph $G$ of diameter $2$
does not contain a diametrical triple, then $\theta_3(G) = 4$. The $3$-ASC
index of paths of order $n\geq 1$, cycles of order $n\geq 3$, and trees of
order $n\geq 10$ and diameter $n-2$ are also determined, respectively, and
several open problems proposed.
