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A graph is almost self-centered (ASC) if all but two of its vertices arecentral. An almost self-centered graph with radius $r$ is called an $r$-ASCgraph. The $r$-ASC index $\theta_r(G)$ of a graph $G$ is the minimum number ofvertices needed to be added to $G$ such that an $r$-ASC graph is obtained thatcontains $G$ 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 completegraphs is determined. It is proved that $\theta_3(G)\in \{3,4\}$ if $G$ hasdiameter $2$ and for several classes of graphs of diameter $2$ the exact valueof 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 $3$-ASCindex of paths of order $n\geq 1$, cycles of order $n\geq 3$, and trees oforder $n\geq 10$ and diameter $n-2$ are also determined, respectively, andseveral open problems proposed.

Embeddings into almost self-centered graphs of given radius