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 ), Xu et al. (J Comb Optim 36(4):1388–1410, 2017 ), 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.
Radius Diameter Almost self-centered graphs r-ASC embedding index
This is a preview of subscription content, log in to check access.
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
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
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
Graham, R.L., and P.M. Winkler, On isometric embeddings of graphs. Transactions of the American Mathematical Society 288 (2), 527–536Google Scholar
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
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