The local multiset dimension of graphs


  • Ridho Alfarisi University of Jember
  • Dafik . University of Jember
  • Arika Indah Kristiana University of Jember
  • Ika Hesti Agustin University of Jember





Local Resolving Set, Local Multiset Dimension, Distance, Some Family Graph.


All graphs in this paper are nontrivial and connected graph. For -ordered set  of vertex set , the multiset representation of a vertex  of  with respect to  is  where  is a distance between of the vertex  and the vertices in  together with their multiplicities. The resolving set  is a local resolving set of  if  for every pair  of adjacent vertices of . The minimum local resolving set  is a local multiset basis of . If  has a local multiset basis, then its cardinality is called local multiset dimension,denoted by . If  does not contain a local resolving set, then we write  In our paper, we will investigate the establish sharp bounds of the local multiset dimension of  and determine the exact value of some family graphs.


[1] J. L. Gross, J. Yellen and P. Zhang. {it Handbook of graph Theory} Second Edition CRC Press Taylor and Francis Group, (2014).

[2] G. Chartrand and L. Lesniak. {it Graphs and digraphs} {it 3rd} ed (London: Chapman and Hall), (2000)

[3] N. Hartsfield dan G. Ringel. textit{Pearls in Graph Theory} Academic Press. United Kingdom, (1994).

[4] P.J. Slater, Leaves of trees, in:textit{Proc. 6th Southeast Conf. Comb., Graph Theory, Comput. Boca Rotan}, {bf 14} (1975), 549-559.

[5] F. Harary and R.A. Melter, on the metric dimension of a graph, textit{Ars Combin}, {bf 2} (1976), 191-195.

[6] E. Ulfianita, N. Estuningsih, L. Susilowati. DimensiMetrikLokal pada Graf Hasil Kali Comb dari Graf Siklus dan Graf Lintasan. Journal of Mathematics, Vol. 1 (2014) No. 3, 24 - 33.

[7] F. Okamoto, B. Phinezy, P. Zhang. The Local Metric Dimension of A Graph. Mathematica Bohemica, Vol. 135 (2010) No. 3, 239 - 255

[8] R. Simanjuntak, T. Vetrik, and P. B. Mulia. The multiset dimension of graphs. arXiv preprint arXiv:1711.00225, (2017)

[9] Chartrand, G., Eroh, L., Johnson, M.A., amdOellermann, O.R., 2000, Resolvability in Graphs and the Metric Dimension of a Graph, Discrete Appl. Math., 105: 99-113.

[10] Dafik, Agustin I H, Surahmat, Syafrizal Sy and Alfarisi R 2017 On non-isolated resolving number of some graph operations {it Far East Journal of Mathematical Sciences} {bf 102} (2) 2473 – 2492.

[11] Darmaji, and Alfarisi, R. "On the partition dimension of comb product of path and complete graph." AIP Conference Proceedings. Vol. 1867. No. 1. AIP Publishing, 2017.

[12] Alfarisi, R., Darmaji, and Dafik. "On the star partition dimension of comb product of cycle and complete graph." Journal of Physics Conference Series. Vol. 855. No. 1. 2017.

[13] Alfarisi, Ridho, and Darmaji. "On the star partition dimension of comb product of cycle and path." AIP Conference Proceedings. Vol. 1867. No. 1. AIP Publishing, 2017.

View Full Article: