Metric Dimension, Minimal Doubly Resolving Sets, and the Strong Metric Dimension for Jellyfish Graph and Cocktail Party Graph

Let Gamma be a simple connected undirected graph with vertex set V(Gamma) and edge set E(Gamma). The metric dimension of a graph Gamma is the least number of vertices in a set with the property that the list of distances from any vertex to those in the set uniquely identifies that vertex. For an ordered subset W = {w(1), w(2),..., w(k)} of vertices in a graph Gamma and a vertex v of Gamma, the metric representation of v with respect toWis the k-vector r(v vertical bar W). (d(v, w(1)), d(v, w(2)),..., d(v, w(k))). If every pair of distinct vertices of Gamma have different metric representations, then the ordered set W is called a resolving set of G. It is known that the problem of computing this invariant is NP-hard. In this paper, we consider the problem of determining the cardinality psi(Gamma) of minimal doubly resolving sets of Gamma and the strong metric dimension for the jellyfish graph JFG( n, m) and the cocktail party graph CP(k + 1).