# The local multiset dimension of graphs

## Authors

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

2019-08-25

## Keywords:

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

## Abstract

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.

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