Geodesic
On the Metric Dimension of Circulant Graphs with $2$ Generators
Kragujevac Journal of Mathematics, Tome 43 (2019) no. 1, p. 49 Cet article a éte moissonné depuis la source eLibrary of Mathematical Institute of the Serbian Academy of Sciences and Arts

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A set of vertices $W$ resolves a connected graph $G$ if every vertex of $G$ is uniquely determined by its vector of distances to the vertices in $W$. The~number of vertices in a smallest resolving set is called the metric dimension and it is denoted by $\dim(G)$. We study the~circulant graphs $C_n (2,3)$ with the vertices $v_0, v_1, v_2,\dots, v_{n-1}$ and the edges $v_i v_{i+2}, v_i v_{i+3}$, where $i = 0, 1, 2,\dots, n-1$, the indices are taken modulo $n$. We show that for $n \ge 26$ we have $\dim(C_n (2,3)) = 3$ if $n \equiv 4 \pmod 6$, $\dim(C_n (2,3)) = 4$ if $n \equiv 0, 1, 5 \pmod 6$ and $3 \le \dim (C_n(2,3)) \le 4$ if $n \equiv 2, 3 \pmod 6$.
Classification : 05C35 05C12
Keywords: Metric dimension, resolving set, circulant graph, distance 
@article{KJM_2019_43_1_a4,
     author = {L. du Toit and T. Vetr{\'\i}k},
     title = {On the {Metric} {Dimension} of {Circulant} {Graphs} with $2$ {Generators}},
     journal = {Kragujevac Journal of Mathematics},
     pages = {49 },
     year = {2019},
     volume = {43},
     number = {1},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/KJM_2019_43_1_a4/}
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L. du Toit; T. Vetrík. On the Metric Dimension of Circulant Graphs with $2$ Generators. Kragujevac Journal of Mathematics, Tome 43 (2019) no. 1, p. 49 . http://geodesic.mathdoc.fr/item/KJM_2019_43_1_a4/