Abstract
For an ordered non-empty subset \( S=\{v_1,\ldots , v_k\}\) of vertices in a connected graph G and an l-clique \(V'\) of G, the l-clique metric S-representation of \(V'\) is the vector \(r^l_G(V'|S) = (d_G(V',v_1), \ldots , d_G(V',v_k))\,\) where \(d_G(V',v_i)=\min \{d_G(v,v_i): v\in V'\}\). A non-empty subset S of V(G) is an l-clique metric generator for G if all l-cliques of G have pairwise different l-clique metric S-representations. An l-clique metric generator of smallest order is an l-clique metric basis for G, its order being the l-clique metric dimension (l-CMD for short) \(\mathrm{cdim}_l(G)\) of G. In this paper, we propose this concept as an extension of the 1-clique metric dimension which is known as the metric dimension, and also study some its properties. Moreover, l-CMD for \(\Gamma ({\mathbb {Z}}_n)\) and the corona product of two graphs is investigated. Furthermore, we prove that computing the l-CMD of connected graphs is NP-hard and present an integer linear programming model for finding this parameter.
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Afkhami, M., Barati, Z., Khashyarmanesh, K.: On the Laplacian spectrum of the comaximal graphs (submitted)
Bondy, J.A., Murty, U.S.R.: Graph theory, Graduate Texts in Mathematics, vol. 244. Springer, New York (2008)
Buszkowski, P.S., Chartrand, G., Poisson, C., Zhang, P.: On \(K\)-dimensional graphs and their bases. Periodico Mathematica Hungarica 46, 9–15 (2003)
Caceres, J., Hernando, C., Mora, M., Pelayo, I.M., Puertas, M.L., Seara, C., Wood, D.R.: On the metric dimension of some families of graphs. Electron. Notes Discret. Math. 22, 129–133 (2005)
Caceres, J., Hernando, C., Mora, M., Pelayo, I.M., Puertas, M.L., Seara, C., Wood, D.R.: On the metric dimension of cartesian products of graphs. SIAM J. Discret. Math. 21(2), 423–441 (2007)
Chartrand, G., Eroh, L., Johnson, M.A., Oellermann, O.R.: Resolvability in graphs and the metric dimension of a graph. Discret. Appl. Math. 105, 99–113 (2000)
Cormen, T.H., Leiserson, C.E., Rivest, R.L., Stein, C.: Introduction to Algorithms, 2nd edn. McGraw-Hill book company, The MIT Press (2003)
Cormen, T.H., Leiserson, C.E., Rivest, R.L., Stein, C.: Introduction to Algorithms, 3rd edn. The MIT Press, Cambridge (2009)
Das, K.C., Tavakoli, M.: Bounds for metric dimension and defensive \(k\)-alliance of graphs under deleted lexicographic product. Trans. Comb. 9(1), 31–39 (2020)
Epstein, L.L., Levin, A., Woeginger, G.J.: The (weighted) metric dimension of graphs: hard and easy cases. Algorithmica 72, 1130–1171 (2015)
Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W.H. Freeman, New York (1979)
Hakanen, A., Laihonen, T.: On {l}-metric dimensions in graphs. Fund. Inform. 162, 143–160 (2018)
Johnson, M.: Structure-activity maps for visualizing the graph variables arising in drug design. J. Biopharm. Stat. 3, 203–236 (1993)
Kelenc, A., Tratnik, N., Yero, I.G.: Uniquely identifying the edges of a graph: the edge metric dimension. Discret. Appl. Math. 256, 204–220 (2018)
Khuller, S., Raghavachari, B., Rosenfeld, A.: Landmarks in graphs. Discret. Appl. Math. 70, 217–229 (1996)
Maimani, H.R., Salimi, M., Sattari, A., Yassemi, S.: Comaximal graph of commutative rings. J. Algebra 319, 1801–1808 (2008)
Schwenk, A.J.: Computing the characteristic polynomial of a graph. In: Graphs and Combinatorics. Lecture Notes in Math., vol. 406, pp. 153–172. Springer, Berlin (1974)
Peterin, I., Yero, I.G.: Edge metric dimension of some graph operations. Bull. Malays. Math. Sci. Soc. (2019). https://doi.org/10.1007/s40840-019-00816-7
Saputro, S.W., Simanjuntak, R., Uttunggadewa, S., Assiyatun, H., Baskoro, E.T., Salman, A.N.M., Bača, M.: The metric dimension of the lexicographic product of graphs. Discret. Math. 313, 1045–1051 (2013)
Sharma, P.K., Bhatwadekar, S.M.: A note on graphical representation of rings. J. Algebra 176, 124–127 (1995)
Slater, P.J.: Leaves of trees. Congr. Numer. 14, 549–559 (1975)
Slavko, M.M., Petrovic, Z.Z.: On the structure of comaximal graphs of commutative rings with identity. Bull. Aust. Math. Soc. 83, 11–21 (2011)
Tavakoli, M., Rahbarnia, F., Ashrafi, A.R.: Distribution of some graph invariants over hierarchical product of graphs. Appl. Math. Comput. 220, 405–413 (2013)
Vukičević, D., Zhao, S., Sedlar, J., Xu, S.-J., Došlić, T.: Global forcing number for maximal matchings. Discret. Math. 341, 801–809 (2018)
Wang, H.J.: Graphs associated to co-maximal ideals of commutative rings. J. Algebra 320, 2917–2933 (2008)
Yero, I.G., Kuziak, D., Rodríguez-Velázquez, J.A.: On the metric dimension of corona product graphs. Comput. Math. Appl. 61, 2793–2798 (2011)
Young, M.: Adjacency matrices of zero-divisor graphs of integers modulo \(n\). Involve 8, 753–761 (2015)
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Communicated by Sanming Zhou.
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Afkhami, M., Khashyarmanesh, K. & Tavakoli, M. l-Clique Metric Dimension of Graphs. Bull. Malays. Math. Sci. Soc. 45, 2865–2883 (2022). https://doi.org/10.1007/s40840-022-01299-9
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DOI: https://doi.org/10.1007/s40840-022-01299-9