Partition dimension and strong metric dimension of chain cycle

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.