Research Article
Graphs Associated with the Ideals of a Numerical Semigroup
Having Metric Dimension 2
Ying Wang,
1,2
Muhammad Ahsan Binyamin ,
3
Wajid Ali,
3
Adnan Aslam ,
4
and Yongsheng Rao
2
1
Department of Network Technology, South China Institute of Software Engineering, Guangzhou, China
2
Institute of Computing Science and Technology, Guangzhou University, Guangzhou 510006, China
3
Department of Mathematics, GC University, Faisalabad, Pakistan
4
Department of Natural Sciences and Humanities, University of Engineering and Technology, Lahore (RCET), Pakistan
Correspondence should be addressed to Muhammad Ahsan Binyamin; ahsanbanyamin@gmail.com
Received 18 December 2020; Revised 7 January 2021; Accepted 23 January 2021; Published 9 February 2021
Academic Editor: Ali Ahmad
Copyright © 2021 Ying Wang et al. is is an open access article distributed under the Creative Commons Attribution License,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Let Λ be a numerical semigroup and I ⊂Λ be an irreducible ideal of Λ. e graph G
I
(Λ) assigned to an ideal I of Λ is a graph with
elements of (Λ\I)
∗
as vertices, and any two vertices x and y are adjacent if and only if x + y ∈ I. In this work, we give a complete
characterization (up to isomorphism) of the graph G
I
(Λ) having metric dimension 2.
1. Introduction
In algebraic combinatorics, the study of graphs associated
with algebraic objects is one of the most important and
fascinating fields of research. During the last couple of
decades, a lot of research is carried out in this field. ere are
many papers on assigning graphs to rings, groups, and
semigroups [1–6]. Several authors [7–13] studied different
properties of these graphs including diameter, girth, dom-
ination, metric dimension, central sets, and planarity.
We start by defining some basic concept related to graph
theory. A graph G �(V(G),E(G)) hasavertexset V(G) and
the edge set E(G). e cardinality of the vertex set and edge
set is called the order and size of G, respectively. A path in G
is a sequence of edges u
1
u
2
,u
2
u
3
, ... ,u
k−1
u
k
. A graph G is
connected if every pair of vertices x, y ∈ V(G) is connected
by a path. e distance between two vertices x, y ∈ V(G) is
denoted by d(x, y) and is the length of the shortest path
between them. e diameter of G is denoted by d(G) and is
defined as the largest distance between the vertices of G. Let
U � u
1
,u
2
, ... ,u
r
be an ordered subset of V(G). en, the
r−tuple (d(u, u
1
),d(u, u
2
), ... ,d(u, u
r
)) is the representa-
tion u with respect to U. e vertex u is said to be resolved by
U if (d(u,u
1
),d(u,u
2
), ... ,d(u,u
r
) ≠ (d(v,u
1
),d(v,u
2
), ... ,
d(v,u
r
))) for any vertex v ∈ V(G). e set U is called re-
solving set of G if distinct vertices of G have distinct rep-
resentations with respect to U, and it is called basis of G if it
is a resolving set with minimal cardinality. e metric di-
mension of G, denoted by μ(G), is the cardinality of basis.
e concept of metric dimension was introduced by Slater
[14] and later studied by Harary and Melter [15]. It has many
applications, for example, robot navigation [16], pharma-
ceutical chemistry [17, 18], sonar and coast guard long range
navigation [14], and combinatorial optimization [19].
Let N be set of nonnegative integers. A subset Λ⊂ N is
said to be numerical semigroup if the following holds:
(1) 0 ∈Λ
(2) x + y ∈Λ for all x, y ∈Λ
(3) N\Λ is finite
It is easy to observe that the numerical semigroup is a
commutative monoid. us, the set of numerical semigroups
classifies the set of all submonoids of (N, +). e elements of
the set N\Λ are called gaps of Λ, and the largest element of
this set is known as Frobenius number. Note that every
numerical semigroup is finitely generated; that is, there exist
a set A � a
1
,a
2
, ... ,a
t
such that Λ �〈A〉� n
1
a
1
+ ...
Hindawi
Mathematical Problems in Engineering
Volume 2021, Article ID 6697980, 6 pages
https://doi.org/10.1155/2021/6697980