Geodesic
Characterization of $n$-vertex graphs with metric dimension $n-3$
Mathematica Bohemica, Tome 139 (2014) no. 1, pp. 1-23

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For an ordered set $W=\{w_1,w_2,\ldots ,w_k\}$ of vertices and a vertex $v$ in a connected graph $G$, the ordered $k$-vector $r(v|W):=(d(v,w_1),d(v,w_2),\ldots ,d(v,w_k))$ is called the metric representation of $v$ with respect to $W$, where $d(x,y)$ is the distance between vertices $x$ and $y$. A set $W$ is called a resolving set for $G$ if distinct vertices of $G$ have distinct representations with respect to $W$. The minimum cardinality of a resolving set for $G$ is its metric dimension. In this paper, we characterize all graphs of order $n$ with metric dimension $n-3$.
For an ordered set $W=\{w_1,w_2,\ldots ,w_k\}$ of vertices and a vertex $v$ in a connected graph $G$, the ordered $k$-vector $r(v|W):=(d(v,w_1),d(v,w_2),\ldots ,d(v,w_k))$ is called the metric representation of $v$ with respect to $W$, where $d(x,y)$ is the distance between vertices $x$ and $y$. A set $W$ is called a resolving set for $G$ if distinct vertices of $G$ have distinct representations with respect to $W$. The minimum cardinality of a resolving set for $G$ is its metric dimension. In this paper, we characterize all graphs of order $n$ with metric dimension $n-3$.
DOI : 10.21136/MB.2014.143632
Classification : 05C12
Keywords: resolving set; basis; metric dimension 
Jannesari, Mohsen; Omoomi, Behnaz. Characterization of $n$-vertex graphs with metric dimension $n-3$. Mathematica Bohemica, Tome 139 (2014) no. 1, pp. 1-23. doi: 10.21136/MB.2014.143632
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