The Local Multiset Dimension of Graphs

Abstract

All graphs in this paper are nontrivial and connected graph. For 𝑘-ordered set 𝑊 = {𝑠1, 𝑠2, … , 𝑠𝑘} of vertex set 𝐺, the multiset representation of a vertex 𝑣 of 𝐺 with respect to 𝑊 is 𝑟𝑚(𝑣|𝑊) = {𝑑(𝑣, 𝑠1), 𝑑(𝑣, 𝑠2), … , 𝑑(𝑣, 𝑠𝑘)} where 𝑑(𝑣, 𝑠𝑖) is a distance between of the vertex 𝑣 and the vertices in 𝑊 together with their multiplicities. The resolving set 𝑊 is a local resolving set of 𝐺 if𝑟𝑚(𝑣|𝑊) ≠ 𝑟𝑚(𝑢|𝑊) for every pair 𝑢, 𝑣 of adjacent vertices of 𝐺. The minimum local resolving set 𝑊 is a local multiset basis of 𝐺. If 𝐺 has a local multiset basis, then its cardinality is called local multiset dimension,denoted by 𝜇𝑙(𝐺). If 𝐺 does not contain a local resolving set, then we write 𝜇𝑙(𝐺) = ∞. In our paper, we will investigate the establish sharp bounds of the local multiset dimension of 𝐺 and determine the exact value of some family graphs.