The true twin classes-based investigation for connected local dimensions of connected graphs

Let $G$ be a connected graph of order $n$. The representation of a vertex $v$ of $G$ with respect to an ordered set $W=\{w_1,w_2,...,w_k\}$ is the $k$-vector $r(v|W)=(d(v,w_1),d(v,w_2),...,d(v,w_k))$, where $d(v,w_i)$ represents the distance between vertices $v$ and $w_i$ for $1\le i\le k$. An ordered set $W$ is called a connected local resolving set of $G$ if distinct adjacent vertices have distinct representations with respect to $W$, and the subgraph $⟨W⟩$ induced by $W$ is connected. A connected local resolving set of $G$ of minimum cardinality is a connected local basis of $G$, and this cardinality is the connected local dimension $\text{cld}(G)$ of $G$. Two vertices $u$ and $v$ of $G$ are true twins if $N[u]=N[v]$. In this paper, we establish a fundamental property of a connected local basis of a connected graph $G$. We analyze the connected local dimension of a connected graph without a singleton true twin class and explore cases involving singleton true twin classes. Our investigation reveals that a graph of order $n$ contains at most two non-singleton true twin classes when $\text{cld}(G)=n-2$. Essentially, our work contributes to the characterization of graphs with a connected local dimension of $n-2$.