# Embeddings into almost self-centered graphs of given radius

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## Abstract

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 \(r\ge 3\) 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 \(n\ge 1\), cycles of order \(n\ge 3\), and trees of order \(n\ge 10\) and diameter \(n-2\) are also determined, respectively, and several open problems proposed.

## Keywords

Eccentricity Diameter Almost self-centered graph Graph of diameter 2 ASC index## Mathematics Subject Classification

05C12 05C05 05C75## Notes

### Acknowledgements

Kexiang Xu is supported by NNSF of China (No. 11671202), China Postdoctoral Science Foundation (2013M530253, 2014T70512). Kinkar Ch. Das is supported by National Research Foundation funded by the Korean government with the Grant No. 2017R1D1A1B03028642. Sandi Klavžar acknowledges the financial support from the Slovenian Research Agency (research core funding No. P1-0297). We are much grateful to two anonymous referees for their careful reading and helpful comments on our paper which have greatly improved the original version of this paper.

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