TY - JOUR

T1 - On The Local Multiset Dimension of some Families of Graphs

AU - Alfarisi, Ridho

AU - Susilowati, Liliek

AU - Dafik,

AU - Fadekemi, Osaye J.

N1 - Funding Information:
We gratefully acknowledge the support from Universitas Jember and Universitas Airlangga for the year 2023.
Funding Information:
This research was supported by Universitas Airlangga and University of Jember.
Publisher Copyright:
© 2023 World Scientific and Engineering Academy and Society. All rights reserved.

PY - 2023

Y1 - 2023

N2 - Let G be a connected graph. G is said to be unicyclic if it contains exactly one cycle, and bicyclic if the number of edges equals the number of vertices plus one. For a k-ordered set W = {s1, s2, , sk} ⊂ V(G), the multiset representation of a vertex x in G with respect to W is given as rm(x|W) = {d(x, s1), d(x, s2), , d(x, sk)}, where d(x, si) is the distance between x and the ordered subset si of W together with their multiplicities. The set W is called a local m-resolving set of G if for every uν ∈ E(G), rm(u|W) ≠ rm(ν|W). The local m-resolving set with minimum cardinality is called the local multiset basis and its cardinality is called the local multiset dimension of G, denoted by mdl (G). If G has no local m- resolving set, we write mdt (G) = ∞ and say that G has an infinite local multiset dimension. In this paper, we determine the local multiset dimension of the unicyclic and bicyclic graphs.

AB - Let G be a connected graph. G is said to be unicyclic if it contains exactly one cycle, and bicyclic if the number of edges equals the number of vertices plus one. For a k-ordered set W = {s1, s2, , sk} ⊂ V(G), the multiset representation of a vertex x in G with respect to W is given as rm(x|W) = {d(x, s1), d(x, s2), , d(x, sk)}, where d(x, si) is the distance between x and the ordered subset si of W together with their multiplicities. The set W is called a local m-resolving set of G if for every uν ∈ E(G), rm(u|W) ≠ rm(ν|W). The local m-resolving set with minimum cardinality is called the local multiset basis and its cardinality is called the local multiset dimension of G, denoted by mdl (G). If G has no local m- resolving set, we write mdt (G) = ∞ and say that G has an infinite local multiset dimension. In this paper, we determine the local multiset dimension of the unicyclic and bicyclic graphs.

KW - Networks infrastructure

KW - bicyclic graphs

KW - local m-resolving set

KW - local multiset dimension

KW - unicyclic graphs

UR - http://www.scopus.com/inward/record.url?scp=85147366789&partnerID=8YFLogxK

U2 - 10.37394/23206.2023.22.8

DO - 10.37394/23206.2023.22.8

M3 - Article

AN - SCOPUS:85147366789

SN - 1109-2769

VL - 22

SP - 64

EP - 69

JO - WSEAS Transactions on Mathematics

JF - WSEAS Transactions on Mathematics

ER -