Ambient Communications and Computer Systems pp 181-192 | Cite as

# Almost Self-centered Index of Some Graphs

- 16 Downloads

## Abstract

For a simple connected graph *G*, center *C*(*G*) and periphery *P*(*G*) are subgraphs induced on vertices of *G* with minimum and maximum eccentricity, respectively. An *n*-vertex graph *G* is said to be an almost self-centered (ASC) graph if it contains \(n-2\) central vertices and two peripheral (diametral) vertices. An ASC graph with radius *r* is known as an *r*-ASC graph. The *r*-ASC index of any graph *G* is defined as the minimum number of new vertices, and required edges, to be introduced to *G* such that the resulting graph is *r*-ASC graph in which *G* is induced. For \(r=2,3\), *r*-ASC index of few graphs is calculated by Klavžar et al. (Acta Mathematica Sinica, 27:2343–2350, 2011 [1]), Xu et al. (J Comb Optim 36(4):1388–1410, 2017 [2]), respectively. Here we give bounds to *r*-ASC index of diameter two graphs and determine the exact value of this index for paths and cycles.

## Keywords

Radius Diameter Almost self-centered graphs*r*-ASC embedding index

## References

- 1.Klavžar, S., K.P. Narayankar, and H.B. Walikar. 2011. Almost self-centered graphs.
*Acta Mathematica Sinica*27: 2343–2350.MathSciNetCrossRefGoogle Scholar - 2.Xu, K., H. Liu, K.C. Das, and S. Klavžar. 2017. Embeddings into almost self-centered graphs of given radius.
*Journal of Combinatorial Optimization*36 (4): 1388–1410.MathSciNetCrossRefGoogle Scholar - 3.Balakrishnan, K., B. Bresar, M. Changat, S. Klavzar, M. Kovse, and A. Subhamathi. 2010. Simultaneous embeddings of graphs as median and antimedian subgraphs.
*Networks*56: 90–94.MathSciNetzbMATHGoogle Scholar - 4.Buckley, F., Z. Miller, and P.J. Slater. 1981. On graphs containing a given graph as center.
*Journal of Graph Theory*5: 427–434.MathSciNetCrossRefGoogle Scholar - 5.Dankelmanna, P., and G. Sabidussib. 2008. Embedding graphs as isometric medians.
*Discrete Applied Mathematics*156: 2420–2422.MathSciNetCrossRefGoogle Scholar - 6.Graham, R.L., and P.M. Winkler, On isometric embeddings of graphs.
*Transactions of the American Mathematical Society*288 (2), 527–536Google Scholar - 7.Janakiraman, T.N., M. Bhanumathi, and S. Muthammai. 2008. Self-centered super graph of a graph and center number of a graph.
*Ars Combinatoria*87: 271–290.MathSciNetzbMATHGoogle Scholar - 8.Klavžar, S., H. Liu, P. Singh, and K. Xu. 2017. Constructing almost peripheral and almost self-centered graphs revisited.
*Taiwanese Journal of Mathematics*21 (4): 705–717.MathSciNetCrossRefGoogle Scholar - 9.Ostrand, P.A. 1973. Graphs with specified radius and diameter.
*Discrete Mathematics*4: 71–75.MathSciNetCrossRefGoogle Scholar