Certain Varieties of Resolving Sets of A Graph

Badekara Sooryanarayana, Suma A.S., Chandrakala S.B.


Let G=(V,E) be a simple connected graph. For each ordered subset S={s_1,s_2,...,s_k} of V and a vertex u in V, we associate a vector Gamma(u/S)=(d(u,s_1),d(u,s_2),...,d(u,s_k)) with respect to S, where d(u,v) denote the distance between u and v in G. A subset S is said to be resolving set of G if Gamma(u/S) not equal to Gamma(v/S) for all u, v in V-S. The purpose of this paper is to introduce various types of r-sets and compute minimum cardinality of each set, in possible cases, particulary for paths, cycles, complete graphs and wheels.


simple resolving sets; metric dimensions; landmarks; maximal resolving sets; null resolving sets.

Full Text:



F. Buckley and F. Harary, Distance in graphs, Addison-Wesley, (1990).

J. Caceres, C. Hernando, M. Mora, I.M. Pelayoe, M.L. Puertas, C. Seara, D.R. Wood, On the metric dimension of some families of graphs, Electronic Notes in Discrete Mathematics, 14(22)(2005), 129-133.

Gary Chartrand, Linda Eroh, Mark A. Johnson, and Ortrud R. Oellermann, Resolvability in graphs and the metric dimension of a graph, Discrete Appl. Math., 105(1-3)(2000), 99-113.

Hartseld Gerhard and Ringel, Pearls in graph theory, Academic Press, USA, (1994).

F. Harary and R.A. Melter, On the metric dimension of a graph, Ars. Combinatoria, 2(1976), 19 191-195.

S. Khuller, B. Raghavachari and A. Rosened, Landmarks in graphs, Discrete. Appl. Math., 21(70) (1996), 217-229.

Padma.M.M and Jayalakshmi M, Boundary values of rr; rr;Rr;Rr sets of certain classes of graphs, Theoretical Mathematics and Applications, 7(1) (2017), 29-39.

Padma M M and Jayalakshmi M, On classes of rational resolving sets of derived graphs of a path, Far. East. J. Mathematical Sciences, 110(2) (2019), 247-259.

A. Sebo and E. Tannier, On metric generators of graphs, Math. Opr. Res., 29(2) (2004), 383-393.

B. Shanmukha, B. Sooryanarayana and K.S. Harinath, Metric dimention of wheels, Far East J. Appl. Math., 8(3) (2002), 217-229.

P. J. Slater, Leaves of trees, In Proc. 6th Southeastern Conf. on Combinatorics, Graph Theory, and Computing, Congressus Numerantium,14(1975), 549-559.

B. Sooryanarayana and Shanmukha B, A note on metric dimension, Far East J. Appl. Math., 5(3)(2001), 331-339.

B. Sooryanarayana, On the metric dimension of graph, Indian. J. pure appl. Math., 29(4)(1998), 413-415.

B. Sooryanarayana and Suma A. S, On classes of neighborhood resolving sets of a graph, Electronic Journal of Graph Theory and Applications, 6(1)(2018), 29-36.

B. Sooryanarayana, Shreedhar K and Narahari N, Metric dimension of generalized wheels, Arab J.Math.Sci. 25(2)(2019), 131-144.

Varaporn Saenpholphat and Ping Zhang, Connected resolvability of graphs, Czechoslovak Math. J., 53(128)(4)(2003), 827-840.

DOI: https://doi.org/10.22342/jims.27.1.881.103-114


  • There are currently no refbacks.

Journal of the Indonesian Mathematical Society
Mathematics Department, Universitas Gadjah Mada
Senolowo, Sinduadi, Mlati, Sleman Regency, Special Region of Yogyakarta 55281, Telp. (0274) 552243
Email: jims.indoms@gmail.com

p-ISSN: 2086-8952 | e-ISSN: 2460-0245

Journal of the Indonesian Mathematical Society is licensed under a Creative Commons Attribution 4.0 International License

web statistics
View My Stats