Geodesic
On the Metric Dimension of Directed and Undirected Circulant Graphs
Discussiones Mathematicae. Graph Theory, Tome 40 (2020) no. 1, pp. 67-76

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The undirected circulant graph C_n(±1, ±2, . . ., ±t) consists of vertices v_0, v_1, . . ., v_n−1 and undirected edges v_iv_i+j, where 0 ≤ i ≤ n − 1, 1 ≤ j ≤ t (2 ≤ t ≤ n/2), and the directed circulant graph C_n(1, t) consists of vertices v_0, v_1, . . ., v_n−1 and directed edges v_iv_i+1, v_iv_i+t, where 0 ≤ i ≤ n − 1 (2 ≤ t ≤ n−1), the indices are taken modulo n. Results on the metric dimension of undirected circulant graphs C_n(±1, ±t) are available only for special values of t. We give a complete solution of this problem for directed graphs C_n(1, t) for every t ≥ 2 if n ≥ 2t^2. Grigorious et al. [On the metric dimension of circulant and Harary graphs, Appl. Math. Comput. 248 (2014) 47–54] presented a conjecture saying that dim (C_n(±1, ±2, . . ., ±t)) = t + p − 1 for n = 2tk + t + p, where 3 ≤ p ≤ t + 1. We disprove it by showing that dim (C_n(±1, ±2, . . ., ±t)) ≤ t + p+1/2 for n = 2tk + t + p, where t ≥ 4 is even, p is odd, 1 ≤ p ≤ t + 1 and k ≥ 1.
Keywords: metric dimension, resolving set, circulant graph, distance 
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Vetrík, Tomáš. On the Metric Dimension of Directed and Undirected Circulant Graphs. Discussiones Mathematicae. Graph Theory, Tome 40 (2020) no. 1, pp. 67-76. http://geodesic.mathdoc.fr/item/DMGT_2020_40_1_a4/