Czechoslovak Mathematical Journal, Vol. 69, No. 3, pp. 621-636, 2019


Resolving sets of directed Cayley graphs for the direct product of cyclic groups

Demelash Ashagrie Mengesha, Tomáš Vetrík

Received March 18, 2017.   Published online May 17, 2019.

Abstract:  A directed Cayley graph $C(\Gamma,X)$ is specified by a group $\Gamma$ and an identity-free generating set $X$ for this group. Vertices of $C(\Gamma,X)$ are elements of $\Gamma$ and there is a directed edge from the vertex $u$ to the vertex $v$ in $C(\Gamma,X)$ if and only if there is a generator $x \in X$ such that $ux = v$. We study graphs $C(\Gamma,X)$ for the direct product $Z_m \times Z_n$ of two cyclic groups $Z_m$ and $Z_n$, and the generating set $X = \{ (0,1), (1, 0), (2,0), \dots, (p,0) \}$. We present resolving sets which yield upper bounds on the metric dimension of these graphs for $p = 2$ and $3$.
Keywords:  metric dimension; resolving set; Cayley graph; direct product; cyclic group
Classification MSC:  05C25, 05C12
DOI:  10.21136/CMJ.2019.0127-17


References:
[1] A. Ahmad, M. Imran, O. Al-Mushayt, S. A. U. H. Bokhary: On the metric dimension of barycentric subdivision of Cayley graphs Cay$(Z_n \oplus Z_m)$. Miskolc Math. Notes 16 (2016), 637-646. DOI 10.18514/MMN.2015.1192 | MR 3454129 | Zbl 1349.05082
[2] G. Chartrand, L. Eroh, M. A. Johnson, O. R. Oellermann: Resolvability in graphs and the metric dimension of a graph. Discrete Appl. Math. 105 (2000), 99-113. DOI 10.1016/S0166-218X(00)00198-0 | MR 1780464 | Zbl 0958.05042
[3] M. Fehr, S. Gosselin, O. R. Oellermann: The metric dimension of Cayley digraphs. Discrete Math. 306 (2006), 31-41. DOI 10.1016/j.disc.2005.09.015 | MR 2202072 | Zbl 1085.05034
[4] F. Harary, R. A. Melter: On the metric dimension of a graph. Ars Comb. 2 (1976), 191-195. MR 0457289 | Zbl 0349.05118
[5] M. Imran: On the metric dimension of barycentric subdivision of Cayley graphs. Acta Math. Appl. Sin. Engl. Ser. 32 (2016), 1067-1072. DOI 10.1007/s10255-016-0627-0 | MR 3552871 | Zbl 06700451
[6] M. Imran, A. Q. Baig, S. A. U. H. Bokhary, I. Javaid: On the metric dimension of circulant graphs. Appl. Math. Lett. 25 (2012), 320-325. DOI 10.1016/j.aml.2011.09.008 | MR 2855980 | Zbl 1243.05072
[7] I. Javaid, M. T. Rahim, K. Ali: Families of regular graphs with constant metric dimension. Util. Math. 75 (2008), 21-33. MR 2389696 | Zbl 1178.05037
[8] S. Khuller, B. Raghavachari, A. Rosenfeld: Landmarks in graphs. Discrete Appl. Math. 70 (1996), 217-229. DOI 10.1016/0166-218X(95)00106-2 | MR 1410574 | Zbl 0865.68090
[9] R. A. Melter, I. Tomescu: Metric bases in digital geometry. Comput. Vision Graphics Image Process 25 (1984), 113-121. DOI 10.1016/0734-189X(84)90051-3 | Zbl 0591.51023
[10] O. R. Oellermann, C. D. Pawluck, A. Stokke: The metric dimension of Cayley digraphs of Abelian groups. Ars Comb. 81 (2006), 97-111. MR 2267805 | Zbl 1189.05055
[11] P. J. Slater: Leaves of trees. Proc. 6th Southeast. Conf. Combinatorics, Graph Theory and Computing Congressus Numerantium 14, Utilitas Mathematica, Winnipeg (1975), 549-559. MR 0422062 | Zbl 0316.05102

Affiliations:   Demelash Ashagrie Mengesha, Tomáš Vetrík, Department of Mathematics and Applied Mathematics, University of the Free State, P. O. Box 339, Bloemfontein, 9300, Free State, South Africa, e-mail: 2014215681@ufs4life.ac.za, vetrikt@ufs.ac.za


 
PDF available at: