Embeddings into almost self-centered graphs of given radius

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 $$\theta _r(G)$$θr(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 $$\theta _r(G)\le 2r$$θr(G)≤2r holds for any graph G and any $$r\ge 2$$r≥2 which improves the earlier known bound $$\theta _r(G)\le 2r+1$$θr(G)≤2r+1. It is further proved that $$\theta _r(G)\le 2r-1$$θr(G)≤2r-1 holds if $$r\ge 3$$r≥3 and G is of order at least 2. The 3-ASC index of complete graphs is determined. It is proved that $$\theta _3(G)\in \{3,4\}$$θ3(G)∈{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 $$\theta _3(G) = 4$$θ3(G)=4. The 3-ASC index of paths of order $$n\ge 1$$n≥1, cycles of order $$n\ge 3$$n≥3, and trees of order $$n\ge 10$$n≥10 and diameter $$n-2$$n-2 are also determined, respectively, and several open problems proposed.