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

Abstract

Let Γ be a simple connected undirected graph with vertex set VΓ and edge set EΓ. The metric dimension of a graph Γ 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=w1,w2,…,wk of vertices in a graph Γ and a vertex v of Γ, the metric representation of v with respect to W is the k-vector rvW=dv,w1,dv,w2,…,dv,wk. If every pair of distinct vertices of Γ have different metric representations, then the ordered set W is called a resolving set of Γ. It is known that the problem of computing this invariant is NP-hard. In this paper, we consider the problem of determining the cardinality ψΓ of minimal doubly resolving sets of Γ and the strong metric dimension for the jellyfish graph JFGn,m and the cocktail party graph CPk+1.

Computing Edge Version of Resolvability and Double Resolvability of a Graph

“…In [30], Pan et al investigated the doubly resolving number for various types of convex polytopes. Liu et al [31] studied the MD and minimal DRSs for the cocktail party graphs and jellyfish graphs. Further details are discussed in [32,33].…”

Computing Minimal Doubly Resolving Sets and the Strong Metric Dimension of the Layer Sun Graph and the Line Graph of the Layer Sun Graph

“…In this paper, we have constructed a layer Sun graph LSG(n, m, k), discussed this graph, and computed the minimum cardinality of the doubly resolving set and strong resolving set of layer Sun graph LSG(n, m, k) and the line graph of the layer Sun graph LSG(n, m, k). We deduce that, by this way, we can construct a layer jellyfish graph JFG(n, m, k), of order n + Σ k−1 r�1 nm r , where the jellyfish graph is JFG(n, m), which is defined in [5], and by a similar way, we can obtain and compute the minimum cardinality of doubly resolving set and strong resolving set of layer jellyfish graph JFG(n, m, k) and the line graph of the layer jellyfish graph JFG(n, m, k).…”

“…e notion of a strong metric dimension problem set of vertices of the graph G was introduced by A. Sebö and E. Tannier [3] and further investigated by O. R. Oellermann and Peters-Fransen [4]. e minimal doubly resolving sets for jellyfish and cocktail party graphs have been obtained in [5]. For more results related to these concepts, see [6][7][8][9][10][11][12][13][14][15][16].…”

Some Resolving Parameters in a Class of Cayley Graphs

“…Also, for convenience, we can use the symbols in the Cayley graph Λ � Cay(D 2n , Ψ), instead of the symbols in the Toeplitz graph T 2n (W). Some metrics for a class of distance regular graphs is computed in [13,14]. On the contrary, because of the difficulty of the computing resolving parameters of a class of graphs which are not distance regular, we regard this as justification for our focus on some resolving parameters in the Cayley graph Λ � Cay(D 2n , Ψ).…”