Partition dimension and strong metric dimension of chain cycle

TL;DR

This paper investigates the partition dimension and strong metric dimension of chain cycle graphs, providing exact values for those constructed from even and odd cycles, advancing understanding of graph resolving parameters.

Contribution

It determines the partition dimension and strong metric dimension specifically for chain cycle graphs built from even and odd cycles, which was previously unexplored.

Findings

Partition dimension of chain cycles from even cycles is established.

Strong metric dimension of chain cycles from odd cycles is calculated.

Provides formulas for dimensions based on cycle composition.

Abstract

Let $G$ be a connected graph with vertex set $V(G)$ and edge set $E(G)$. For an ordered $k$-partition $\Pi=\{Q_1,\ldots,Q_k\}$ of $V(G)$, the representation of a vertex $v \in V(G)$ with respect to $\Pi$ is the $k$-vectors $r(v|\Pi)=(d(v,Q_1),\ldots,d(v,Q_k))$, where $d(v,Q_i)$ is the distance between $v$ and $Q_i$. The partition $\Pi$ is a resolving partition if $r(u|\Pi)\neq r(v|\Pi)$, for each pair of distinct vertices $u,v \in V(G)$. The minimum $k$ for which there is a resolving $k$-partition of $V(G)$ is the partition dimension of $G$. A vertex $w\in V(G)$ strongly resolves two distinct vertices $u,v \in V(G)$ if $u$ belongs to a shortest $v-w$ path or $v$ belongs to a shortest $u-w$ path. An ordered set $W=\{w_{1},\ldots, w_{t}\}\subseteq V(G)$ is a strong resolving set for $G$ if for every two distinct vertices $u$ and $v$ of $G$ there exists a vertex $w\in W$ which strongly resolves $u$ and $v$. A strong metric basis of $G$ is a strong resolving set of minimal cardinality. The cardinality of a strong metric basis is called strong metric dimension of $G$. In this paper, we determine the partition dimension and strong metric dimension of a chain cycle constructed by even cycles and a chain cycle constructed by odd cycles.