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Two vertices u and v in a nontrivial connected graph G are twins if u and v have the same neighbors in V (G) − {u, v}. If u and v are adjacent, they are referred to as true twins; while if u and v are nonadjacent, they are false twins. For a positive integer k, let c : V (G) → Zk be a vertex coloring where adjacent vertices may be assigned the same color. The coloring c induces another vertex coloring c′ : V (G) → Zk defined by c′(v) = P u∈N[v] c(u) for each v ∈ V (G), where N[v] is the closed neighborhood...

For an ordered set $W=\{w_1,w_2,...,w_k\}$ of $k$ distinct vertices in a nontrivial connected graph $G$, the metric code of a vertex $v$ of $G$ with respect to $W$ is the $k$-vector $$\mathrm{code}\left(v\right)=(d(v,w_1),d(v,w_2),\cdots ,d(v,w_k))$$ where $d(v,w_i)$ is the distance between $v$ and $w_i$ for $1\le i\le k$. The set $W$ is a local metric set of $G$ if $\mathrm{code}\left(u\right)\ne \mathrm{code}\left(v\right)$ for every pair $u,v$ of adjacent vertices of $G$. The minimum positive integer $k$ for which $G$ has a local metric $k$-set is the local metric dimension $\mathrm{lmd}\left(G\right)$ of $G$. A local metric set of $G$ of cardinality $\mathrm{lmd}\left(G\right)$ is a local metric basis of $G$. We characterize all nontrivial connected...