## 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.

This is a preview of subscription content, log in to check access.

## References

Balakrishnan K, Brešar B, Changat M, Klavžar S, Peterin I, Subhamathi AR (2012) Almost self-centered median and chordal graphs. Taiwan J Math 16:1911–1922

Buckley F (1989) Self-centered graphs, Graph Theory and Its Applications: East and West. Ann N Y Acad Sci 576:71–78

Cheng YK, Kang LY, Yan H (2014) The backup 2-median problem on block graphs. Acta Math Appl Sin Engl Ser 30:309–320

Das KC, Nadjafi-Arani MJ (2017) On maximum Wiener index of trees and graphs with given radius. J Comb Optim 34:574–587

Das KC, Lee DW, Graovac A (2013) Some properties of the Zagreb eccentricity indices. ARS Math Contemp 6:117–125

Diestel R (2006) Graph theory. Springer, Berlin

Gupta S, Singh M (2002) Application of graph theory: relationship of eccentric connectivity index and Wiener’s index with anti-inflammatory activity. J Math Anal Appl 266:259–268

Gupta S, Singh M, Madan AK (2000) Connective eccentricity index: a novel topological descriptor for predicting biological activity. J Mol Graph Model 18:18–25

Gupta S, Singh M, Madan AK (2002) Eccentric distance sum: a novel graph invariant for predicting biological and physical properties. J Math Anal Appl 275:386–401

Hong YM, Kang LY (2012) Backup 2-center on interval graphs. Theor Comput Sci 445:25–35

Huang TC, Lin JC, Chen HJ (2000) A self-stabilizing algorithm which finds a 2-center of a tree. Comput Math Appl 40:607–624

Ilić A, Yu G, Feng L (2011) On the eccentric distance sum of graphs. J Math Anal Appl 381:590–600

Klavžar S, Narayankar KP, Walikar HB (2011) Almost self-centered graphs. Acta Math Sin (Engl Ser) 27:2343–2350

Klavžar S, Narayankar KP, Walikar HB, Lokesh SB (2014) Almost-peripheral graphs. Taiwan J Math 18:463–471

Klavžar S, Liu H, Singh P, Xu K (2017) Constructing almost peripheral and almost self-centered graphs revisited. Taiwan J Math 21:705–717

Krnc M, Škrekovski R (2015) Group centralization of network indices. Discrete Appl Math 186:147–157

Maddaloni A, Zamfirescu CT (2016) A cut locus for finite graphs and the farthest point mapping. Discrete Math 339:354–364

Morgan MJ, Mukwembi S, Swart HC (2011) On the eccentric connectivity index of a graph. Discrete Math 311:1229–1234

Palacios JL (2015) On the Kirchhoff index of graphs with diameter 2. Discrete Appl Math 184:196–201

Puerto J, Tamir A, Mesa JA, Pérez-Brito D (2008) Center location problems on tree graphs with subtree-shaped customers. Discrete Appl Math 156:2890–2910

Sharma V, Goswami R, Madan AK (1997) Eccentric connectivity index: a novel highly discriminating topological descriptor for structure-property and structure-activity studies. J Chem Inf Comput Sci 37:273–282

Su G, Xiong L, Su X, Chen X (2015) Some results on the reciprocal sum-degree distance of graphs. J Comb Optim 30:435–446

Tomescu I (2008) Properties of connected graphs having minimum degree distance. Discrete Math 309:2745–2748

Tomescu I (2010) Ordering connected graphs having small degree distances. Discrete Appl Math 158:1714–1717

Wang HL, Wu B, Chao KM (2009) The backup 2-center and backup 2-median problems on trees. Networks 53:39–49

Wu B, An X, Liu G, Yan G, Liu X (2013) Minimum degree, edge-connectivity and radius. J Comb Optim 26:585–591

Xu K, Liu M, Das KC, Gutman I, Furtula B (2014) A survey on graphs extremal with respect to distance-based topological indices. MATCH Commun Math Comput Chem 71:461–508

Xu K, Das KC, Liu H (2016) Some extremal results on the connective eccentricity index of graphs. J Math Anal Appl 433:803–817

Xu K, Das KC, Maden AD (2016) On a novel eccentricity-based invariant of a graph. Acta Math Sin (Engl Ser) 32:1477–1493

Yu G, Feng L (2013) On the connective eccentricity index of graphs. MATCH Commun Math Comput Chem 69:611–628

Yu G, Qu H, Tang L, Feng L (2014) On the connective eccentricity index of trees and unicyclic graphs with given diameter. J Math Anal Appl 420:1776–1786

## 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.

## Author information

### Affiliations

### Corresponding author

## Rights and permissions

## About this article

### Cite this article

Xu, K., Liu, H., Das, K.C. *et al.* Embeddings into almost self-centered graphs of given radius.
*J Comb Optim* **36, **1388–1410 (2018). https://doi.org/10.1007/s10878-018-0311-9

Published:

Issue Date:

### Keywords

- Eccentricity
- Diameter
- Almost self-centered graph
- Graph of diameter 2
- ASC index

### Mathematics Subject Classification

- 05C12
- 05C05
- 05C75