On metric dimension of cube of trees

Paul, Sanchita; Das, Bapan; Adhikari, Avishek; Saha, Laxman

doi:10.22049/cco.2025.29750.2143

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	On metric dimension of cube of trees
Communications in Combinatorics and Optimization
مقالات آماده انتشار، پذیرفته شده، انتشار آنلاین از تاریخ 09 اسفند 1403 اصل مقاله (543.4 K)
نوع مقاله: Original paper
شناسه دیجیتال (DOI): 10.22049/cco.2025.29750.2143
نویسندگان
Sanchita Paul^* ¹؛ Bapan Das²؛ Avishek Adhikari³؛ Laxman Saha⁴
¹Department of Mathematics, C.V. Raman Global University, Bhubaneswar 752054, India
²Department of Mathematics, Balurghat College, Balurghat 733101, India
³Department of Mathematics, Presidency University, Kolkata 700073, India
⁴Department of Mathematics, Balurghat College, Balurghat 733101, India
چکیده
Let $G=(V,E)$ be a connected graph and $d_{G}(u,v)$ be the shortest distance between the vertices $u$ and $v$ in $G$. A set $S=\{s_{1},s_{2},\dots,s_{n}\}\subset V(G)$ is said to be a {\em resolving set} if for all distinct vertices $u,v$ of $G$, there exists an element $s\in S$ such that $d_{G}(s,u)\neq d_{G}(s,v)$. The minimum cardinality of a resolving set for a graph $G$ is called the metric dimension of $G$, and it is denoted by $\beta{(G)}$. A resolving set having $\beta{(G)}$ number of vertices is named as metric basis of $G$. The metric dimension problem is to find a metric basis in a graph $G$, and it has several real-life applications in network theory, telecommunication, image processing, pattern recognition, and many other fields. In this article, we consider cube of trees $T^{3}=(V, E)$, where any two vertices $u,v$ are adjacent if and only if the distance between them is less than or equal to three in $T$. We establish the necessary and sufficient conditions for a vertex subset of $V$ to become a resolving set for $T^{3}$. This helps to determine the tight bounds (upper and lower) on the metric dimension of $T^{3}$. Then, for certain well-known cube of trees, such as caterpillars, lobsters, spiders, and $d$-regular trees, we establish the boundaries for the metric dimension. Also, for every positive integer, we provide a construction showing the existence of a cube of a tree satisfying its metric dimension as the given integer. Further, we characterize some restricted families of cube of trees satisfying $\beta{(T^{3})}=\beta{(T)}$.
کلیدواژه‌ها
resolving set؛ metric basis؛ metric dimension؛ trees

مراجع
[1] R. Adar and L. Epstein, The weighted 2-metric dimension of trees in the non-landmarks model, Discrete Optim. 17 (2015), 123–135. https://doi.org/10.1016/j.disopt.2015.06.001 [2] M.M. AlHoli, O.A. AbuGhneim, and H.A. Ezeh, Metric dimension of some path related graphs, Global J. Pure Appl. Math. 13 (2017), no. 2, 149–157. [3] M. Ali, M.T. Rahim, and G. Ali, On path related graphs with constant metric dimension, Util. Math. 88 (2012), 203–209. [4] R.F. Bailey and P.J. Cameron, Base size, metric dimension and other invariants of groups and graphs, Bull. Lond. Math. Soc. 43 (2011), no. 2, 209–242. https://doi.org/10.1112/blms/bdq096 [5] R.F. Bailey and I. González Yero, Error-correcting codes from k-resolving sets, Discuss. Math. Graph Theory 39 (2019), no. 2, 341–355. https://doi.org/10.7151/dmgt.2087 [6] Z. Bartha, J. Komjáthy, and J. Raes, Sharp bound on the truncated metric dimension of trees, Discrete Math. 346 (2023), no. 8, Article ID: 113410. https://doi.org/10.1016/j.disc.2023.113410 [7] P. Buczkowski, G. Chartrand, C. Poisson, and P. Zhang, On $k$-dimensional graphs and their bases, Period. Math. Hungar. 46 (2003), no. 1, 9–15. https://doi.org/10.1023/a:1025745406160 [8] G. Chartrand, L. Eroh, M.A. Johnson, and O.R. Oellermann, Resolvability in graphs and the metric dimension of a graph, Discrete Appl. Math. 105 (2000), no. 1-3, 99–113. https://doi.org/10.1016/S0166-218X(00)00198-0 [9] E. Galby, L. Khazaliya, F. Mc Inerney, R. Sharma, and P. Tale, Metric dimension parameterized by feedback vertex set and other structural parameters, SIAM J. Discrete Math. 37 (2023), no. 4, 2241–2264. https://doi.org/10.1137/22M1510911 [10] M.R. Garey and D.S. Johnson, Computers and Intractability: A Guide to the Theory of NP-completeness, wh freeman New York, 2002. [11] A. Hakanen, V. Junnila, T. Laihonen, and I.G. Yero, On vertices contained in all or in no metric basis, Discrete Appl. Math. 319 (2022), 407–423. https://doi.org/10.1016/j.dam.2021.12.004 [12] F. Harary and R.A. Melter, On the metric dimension of a graph, Ars. Combin. 2 (1976), 191–195. [13] I. Javaid, M.T. Rahim, and K. Ali, Families of regular graphs with constant metric dimension, Util. Math. 75 (2008), no. 1, 21–33. [14] S. Khuller, B. Raghavachari, and A. Rosenfeld, Landmarks in graphs, Discrete Appl. Math. 70 (1996), no. 3, 217–229. https://doi.org/10.1016/0166-218X(95)00106-2 [15] M. Knor, R. Škrekovski, and T. Vetrík, Metric dimension of circulant graphs with 5 consecutive generators, Math. 12 (2024), no. 9, Article ID: 1384. https://doi.org/10.3390/math12091384 [16] D. Kuziak and I.G. Yero, Metric dimension related parameters in graphs: A survey on combinatorial, computational and applied results, arXiv preprint arXiv:2107.04877 (2021). [17] S. Mashkaria, G. ´Odor, and P. Thiran, On the robustness of the metric dimension of grid graphs to adding a single edge, Discrete Appl. Math. 316 (2022), 1–27. https://doi.org/10.1016/j.dam.2022.02.014 [18] S. Nawaz, M. Ali, M.A. Khan, and S. Khan, Computing metric dimension of power of total graph, IEEE Access 9 (2021), 74550–74561. https://doi.org/10.1109/ACCESS.2021.3072554 [19] L. Saha, M. Basak, and K. Tiwary, All metric bases and fault-tolerant metric dimension for square of grid, Opuscula Math. 42 (2022), no. 1, 93–111. https://doi.org/10.7494/OpMath.2022.42.1.93 [20] L. Saha, M. Basak, K. Tiwary, K.C. Das, and Y. Shang, On the characterization of a minimal resolving set for power of paths, Math. 10 (2022), no. 14, Article ID: 2445. https://doi.org/10.3390/math10142445 [21] P.J. Slater, Leaves of trees, Congr. Numer. 14 (1975), 549–559. [22] R.C. Tillquist, R.M. Frongillo, and M.E. Lladser, Getting the lay of the land in discrete space: A survey of metric dimension and its applications, SIAM Rev. 65 (2023), no. 4, 919–962. https://doi.org/10.1137/21M1409512 [23] R. Trujillo-Rasua and Ismael G. Yero, k-metric antidimension: A privacy measure for social graphs, Inform. Sci. 328 (2016), 403–417. https://doi.org/10.1016/j.ins.2015.08.048 [24] J. Wu, L. Wang, and W. Yang, Learning to compute the metric dimension of graphs, Appl. Math. Comput. 432 (2022), Article ID: 127350. https://doi.org/10.1016/j.amc.2022.127350 [25] S. Zejnilović, J. Gomes, and B. Sinopoli, Network observability and localization of the source of diffusion based on a subset of nodes, 2013 51st Annual Allerton Conference on Communication, Control, and Computing (Allerton), IEEE, 2013, pp. 847–852.
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On metric dimension of cube of trees