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

Abstract

Let $G$ be a finite, connected graph of order of at least 2, with vertex set $V(G)$ and edge set $E(G)$. A set $S$ of vertices of the graph $G$ is a doubly resolving set for $G$ if every two distinct vertices of $G$ are doubly resolved by some two vertices of $S$. The minimal doubly resolving set of vertices of graph $G$ is a doubly resolving set with minimum cardinality and is denoted by $\psi (G)$. In this paper, first, we construct a class of graphs of order $2n+{\Sigma }_{r=1}^{k-2}n{m}^r$, denoted by $LSG(n,m,k)$, and call these graphs as the layer Sun graphs with parameters $n$, $m$ and $k$. Moreover, we compute minimal doubly resolving sets and the strong metric dimension of layer Sun graph $LSG(n,m,k)$ and the line graph of the layer Sun graph $LSG(n,m,k)$.