On the Connected Partition Dimension of Unicyclic Graphs

Abstract

Let $G$ be a connected graph. For a vertex $v\in V(G)$ and an ordered $k$-partition $\mathrm{\Pi }=(S_1,S_2,\dots ,S_k)$ of $V(G)$, the representation of $v$ with respect to $\mathrm{\Pi }$ is the $k$-vector $r(v|\mathrm{\Pi })=(d(v,S_1),d(v,S_2),\dots ,d(v,S_k))$ where $d(v,S_i)=\underset{w\in S_i}{min}d(x,w)$ ($1\le i\le k$). The $k$-partition $\mathrm{\Pi }$ is said to be resolving if the $k$-vectors $r(v|\mathrm{\Pi })$, $v\in V(G)$, are distinct. The minimum $k$ for which there is a resolving $k$-partition of $V(G)$ is called the partition dimension of $G$, denoted by $pd(G)$. A resolving $k$-partition $\mathrm{\Pi }=\{S_1,S_2,\dots ,S_k\}$ of $V(G)$ is said to be connected if each subgraph $⟨S_i⟩$ induced by $S_i$ ($1\le i\le k$) is connected in $G$. The minimum $k$ for which there is a connected resolving $k$-partition of $V(G)$ is called the connected partition dimension of $G$, denoted by $cpd(G)$. In this paper, the connected partition dimension of the unicyclic graphs is calculated and bounds are proposed.