# 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.

## References

- Balakrishnan K, Brešar B, Changat M, Klavžar S, Peterin I, Subhamathi AR (2012) Almost self-centered median and chordal graphs. Taiwan J Math 16:1911–1922MathSciNetCrossRefGoogle Scholar
- Buckley F (1989) Self-centered graphs, Graph Theory and Its Applications: East and West. Ann N Y Acad Sci 576:71–78CrossRefGoogle Scholar
- Cheng YK, Kang LY, Yan H (2014) The backup 2-median problem on block graphs. Acta Math Appl Sin Engl Ser 30:309–320MathSciNetCrossRefGoogle Scholar
- Das KC, Nadjafi-Arani MJ (2017) On maximum Wiener index of trees and graphs with given radius. J Comb Optim 34:574–587MathSciNetCrossRefGoogle Scholar
- Das KC, Lee DW, Graovac A (2013) Some properties of the Zagreb eccentricity indices. ARS Math Contemp 6:117–125MathSciNetCrossRefGoogle Scholar
- Diestel R (2006) Graph theory. Springer, BerlinzbMATHGoogle Scholar
- 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 266:259–268MathSciNetCrossRefGoogle Scholar
- Gupta S, Singh M, Madan AK (2000) Connective eccentricity index: a novel topological descriptor for predicting biological activity. J Mol Graph Model 18:18–25CrossRefGoogle Scholar
- Gupta S, Singh M, Madan AK (2002) Eccentric distance sum: a novel graph invariant for predicting biological and physical properties. J Math Anal Appl 275:386–401MathSciNetCrossRefGoogle Scholar
- Hong YM, Kang LY (2012) Backup 2-center on interval graphs. Theor Comput Sci 445:25–35MathSciNetCrossRefGoogle Scholar
- Huang TC, Lin JC, Chen HJ (2000) A self-stabilizing algorithm which finds a 2-center of a tree. Comput Math Appl 40:607–624MathSciNetCrossRefGoogle Scholar
- Ilić A, Yu G, Feng L (2011) On the eccentric distance sum of graphs. J Math Anal Appl 381:590–600MathSciNetCrossRefGoogle Scholar
- Klavžar S, Narayankar KP, Walikar HB (2011) Almost self-centered graphs. Acta Math Sin (Engl Ser) 27:2343–2350MathSciNetCrossRefGoogle Scholar
- Klavžar S, Narayankar KP, Walikar HB, Lokesh SB (2014) Almost-peripheral graphs. Taiwan J Math 18:463–471MathSciNetCrossRefGoogle Scholar
- Klavžar S, Liu H, Singh P, Xu K (2017) Constructing almost peripheral and almost self-centered graphs revisited. Taiwan J Math 21:705–717MathSciNetCrossRefGoogle Scholar
- Krnc M, Škrekovski R (2015) Group centralization of network indices. Discrete Appl Math 186:147–157MathSciNetCrossRefGoogle Scholar
- Maddaloni A, Zamfirescu CT (2016) A cut locus for finite graphs and the farthest point mapping. Discrete Math 339:354–364MathSciNetCrossRefGoogle Scholar
- Morgan MJ, Mukwembi S, Swart HC (2011) On the eccentric connectivity index of a graph. Discrete Math 311:1229–1234MathSciNetCrossRefGoogle Scholar
- Palacios JL (2015) On the Kirchhoff index of graphs with diameter 2. Discrete Appl Math 184:196–201MathSciNetCrossRefGoogle Scholar
- Puerto J, Tamir A, Mesa JA, Pérez-Brito D (2008) Center location problems on tree graphs with subtree-shaped customers. Discrete Appl Math 156:2890–2910MathSciNetCrossRefGoogle Scholar
- 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 37:273–282CrossRefGoogle Scholar
- Su G, Xiong L, Su X, Chen X (2015) Some results on the reciprocal sum-degree distance of graphs. J Comb Optim 30:435–446MathSciNetCrossRefGoogle Scholar
- Tomescu I (2008) Properties of connected graphs having minimum degree distance. Discrete Math 309:2745–2748MathSciNetCrossRefGoogle Scholar
- Tomescu I (2010) Ordering connected graphs having small degree distances. Discrete Appl Math 158:1714–1717MathSciNetCrossRefGoogle Scholar
- Wang HL, Wu B, Chao KM (2009) The backup 2-center and backup 2-median problems on trees. Networks 53:39–49MathSciNetCrossRefGoogle Scholar
- Wu B, An X, Liu G, Yan G, Liu X (2013) Minimum degree, edge-connectivity and radius. J Comb Optim 26:585–591MathSciNetCrossRefGoogle Scholar
- 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 71:461–508MathSciNetzbMATHGoogle Scholar
- Xu K, Das KC, Liu H (2016) Some extremal results on the connective eccentricity index of graphs. J Math Anal Appl 433:803–817MathSciNetCrossRefGoogle Scholar
- Xu K, Das KC, Maden AD (2016) On a novel eccentricity-based invariant of a graph. Acta Math Sin (Engl Ser) 32:1477–1493MathSciNetCrossRefGoogle Scholar
- Yu G, Feng L (2013) On the connective eccentricity index of graphs. MATCH Commun Math Comput Chem 69:611–628MathSciNetzbMATHGoogle Scholar
- 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 420:1776–1786MathSciNetCrossRefGoogle Scholar