The forcing metric dimension of a total graph of non-zero annihilating ideals
DOI:
https://doi.org/10.37863/umzh.v75i6.7011Keywords:
forcing metric dimension Zero-divisor; Annihilating ideal, Resolving setAbstract
UDC 519.17
Let R be a commutative ring with identity, which is not an integral domain. An ideal I of a ring R is called an annihilating ideal if there exists r∈R−{0} such that Ir=(0). The total graph of non-zero annihilating ideals of R, denoted by Ω(R), is а graph with the vertex set A(R)∗, the set of all non-zero annihilating ideals of R, and two distinct vertices I and J are joined if and only if I+J is also an annihilating ideal of R. We study the forcing metric dimension of Ω(R) and determine the forcing metric dimension of Ω(R). It is shown that the forcing metric dimension of Ω(R) is equal either to zero or to the metric dimension.
References
N. Abachi, S. Sahebi, On the metric dimension of a total graph of non-zero annihilating ideals, An. Ştiinƫ. Univ. ``Ovidius'' Constanƫa, Ser. Mat., 28, № 3, 5–14 (2020). DOI: https://doi.org/10.2478/auom-2020-0031
M. F. Atiyah, I. G. Macdonald, Introduction to commutative algebra, Addison-Wesley Publ. Co. (1969).
W. Bruns, J. Herzog, Cohen–Macaulay rings, Cambridge Univ. Press (1997). DOI: https://doi.org/10.1017/CBO9780511608681
G. Chartrand, L. Eroh, M. A. Johnson, O. R. Oellermann, Resolvability in graphs and the metric dimension of a graph, Discrete Appl. Math., 105, 99–113 (2000). DOI: https://doi.org/10.1016/S0166-218X(00)00198-0
G. Chartrand, P. Zhang, Kalamazoo, The forcing dimension of a graph, Math. Bohem., 126, № 4, 711–720 (2001). DOI: https://doi.org/10.21136/MB.2001.134116
D. Dolžan, The metric dimension of the annihilating-ideal graph of a finite commutative ring, Bull. Aust. Math. Soc., 103, № 3, 362–368 (2021). DOI: https://doi.org/10.1017/S0004972720001239
D. Dolžan, The metric dimension of the total graph of a finite commutative ring, Canad. Math. Bull., 59, 748–759 (2016). DOI: https://doi.org/10.4153/CMB-2016-015-5
F. Harary, R. A. Melter, On the metric dimension of a graph, Ars Combin., 2, 191–195 (1976).
S. Khuller, B Raghavachari, A. Rosenfeld, Landmarks in graphs, Discrete Appl. Math., 70, № 3, 217–229 (1996). DOI: https://doi.org/10.1016/0166-218X(95)00106-2
T. R. May, O. R. Oellermann, The strong metric dimension of distance hereditary graphs, J. Combin. Math. and Combin. Comput., 76, № 3, 59–73 (2011).
O. R. Oellermann, J. Peters-Fransen, The strong metric dimension of graphs and digraphs, Discrete Appl. Math., 155, 356–364 (2007). DOI: https://doi.org/10.1016/j.dam.2006.06.009
S. Pirzada, M. Aijaz, Metric and upper dimension of zero divisor graphs associated to commutative rings, Acta Univ. Sapientiae Informatica, 12, № 1, 84–101 (2020). DOI: https://doi.org/10.2478/ausi-2020-0006
S. Pirzada, M. Aijaz, S. P. Redmond, Upper dimension and bases of zero divisor graphs of commutative rings, AKCE Int. J. Graphs and Comb., 17, № 1, 168–173 (2019). DOI: https://doi.org/10.1016/j.akcej.2018.12.001
S. Pirzada, M. Imran Bhat, Computing metric dimension of compressed zero divisor graphs associated to rings, Acta Univ. Sapiential Math., 10, № 2, 298–318 (2018). DOI: https://doi.org/10.2478/ausm-2018-0023
B. Shanmukha, B. Sooryanarayana, K. Harinath, Metric dimension of wheels, Far East J. Appl. Math., 8, № 3, 217–229 (2002).
Shikun Ou, Dein Wong, Fenglei Tian, Q. Zhou, Fixing number and metric dimension of a zero-divisor graph associated with a ring, Linear and Multilinear Algebra; DOI/full/10.1080/03081087.2020.1815639 (2020).
S. Visweswaram, H. D. Patel, A graph associated with the set of all nonzero annihilating ideals of a commutative Ring, Discrete Math., Algorithms and Appl., 6, № 4 (2014). DOI: https://doi.org/10.1142/S1793830914500475
D. B. West, Introduction to graph theory, 2nd ed., Prentice Hall, Upper Saddle River (2001).
