Graphs of Neighborhood Metric Dimension Two

Badekara Sooryanarayana (Department of Mathematical and Computational Studies, Dr. Ambedkar Institute of Technology, Bengaluru, Karnataka State, India, Pin 560 056.)
Suma Agani Shanmukha (Department of Mathematics, School of Applied Sciences, REVA University, Yelahanka, Bengaluru, Karnataka State, India, Pin 560 064.)

Journal of Mathematical and Fundamental Sciences
Vol. 53 No. 1 (2021)

A subset  of vertices of a simple connected graph is a neighborhood set (n-set) of  G if G is the union of subgraphs of G induced by the closed neighbors of elements in S. Further, a set S is a resolving set of G if for each pair of distinct vertices x,y of G, there is a vertex s∈ S such that d(s,x)≠d(s,y). An n-set that serves as a resolving set for G is called an nr-set of G. The nr-set with least cardinality is called an nr-metric basis of G and its cardinality is called the neighborhood metric dimension of graph G. In this paper, we characterize graphs of neighborhood metric dimension two.