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

TL;DR

This paper introduces layer Sun graphs, computes their minimal doubly resolving sets and strong metric dimensions, and extends these calculations to their line graphs, advancing understanding of graph resolving parameters.

Contribution

It constructs a new class of graphs called layer Sun graphs and determines their minimal doubly resolving sets and strong metric dimensions, including for their line graphs.

Findings

Computed minimal doubly resolving sets for layer Sun graphs

Determined strong metric dimension of layer Sun graphs

Extended results to line graphs of layer Sun graphs

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 $2n+ \Sigma_{r=1}^{k-2}nm^{r}$, 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 $LSG(n,m, k)$.