On the Edge Metric Dimension of Different Families of Möbius Networks

For an ordered subset Qe of vertices in a simple connected graph G, a vertex x ∈ V distinguishes two edges e1, e2 ∈ E, if d(x, e1)≠d(x, e2). A subset Qe having minimum vertices is called an edge metric generator for G, if every two distinct edges of G are distinguished by some vertex of Qe. 'e minimum cardinality of an edge metric generator for G is called the edge metric dimension, and it is denoted by dime(G). In this paper, we study the edge resolvability parameter for different families of Möbius ladder networks and we find the exact edge metric dimension of triangular, square, and hexagonal Möbius ladder networks.


Introduction
A simple connected graph G � (V, E) with vertex set and edge set V and E, respectively. For two vertices a 1 , a 2 ∈ V, the distance d(a 1 , a 2 ) between vertices a 1 and a 2 is the count of edges between them. A vertex v ∈ V is said to distinguish two vertices a 1 and a 2 , if d(v, a 1 ) ≠ d(v, a 2 ). A set Q ⊂ V is called a resolving set of G, if any pair of distinct vertices of G is distinguished by some element of Q. A resolving set of minimum cardinality is named as metric basis, and its cardinality is the metric dimension of G, denoted by dim(G). A vertex v ∈ V and an edge e � a 1 a 2 ∈ E, and the distance between v and e is defined as d(e, v) � min d(a 1 , v), d(a 2 , v) . A vertex x ∈ V distinguishes two edges e 1 , e 2 ∈ E, if d(x, e 1 ) ≠ d(x, e 2 ). A subset Q e having minimum vertices from a connected graph G is an edge resolving set for G, if every two distinct edges of G are distinguished by some vertex of Q e . e minimum cardinality of an edge resolving set for G is called the edge metric dimension and is denoted by dim e (G).
In 1975, the idea of metric dimension was delivered by Slater [1], he named the metric generators as locating sets which relates to the problem of uniquely recognizing the position of intruders in networks. On the same idea, in 1976 the concept of metric dimension of a graph was independently introduced by Harary and Melter in [2], and these time metric generators were named as resolving sets. One can think that instead of distinguishing two distinct vertices of graph according to chosen subset of vertices, two edges can be distinct with the same subset of vertices. For this concept, Kelenc et al. [3] introduced a new parameter named as the edge metric dimension. In this, they used graph metric to identify each pair of edges by the distance of graph to a chosen subset of vertices.
As far as the idea of metric dimension is extensively studied and used in different fields of science as applications, Chartrand et al. in [4] relate the metric dimension of graph with the drug discovery and pharmacological activity, Khuller et al. in [5] try to put thinking that a robot can be shifted from Euclidean space to graph structure and left his thought as an application of metric dimension in robot navigation, and the metric dimension of Hamming graphs leads Chvátal to the analysis of mastermind games and opens the doors for researchers to view application of metric dimension in complex digital games [6], and in [7,8], Erdös and Lindström assume that the metric dimension can be used in various coin-weighing problems. Resolving sets have served as inspiration for many theoretical studies of graphs. e edge metric dimension is the natural generalization of resolving set, and readers are directed towards the interesting literature containing the metric dimension of different classes of graphs for example, Ali et al. [9,10] studied the metric dimension of Möbius networks and some other cycle-related graphs, Kuziak et al. [11] discussed the strong metric dimension of graphs, Liu et al. [12,13] discussed the metric dimension of cocktail party, Toeplitz, and jellyfish graphs, and some cycle-related graphs were studied with the concept of metric dimension by Ahmad et al. [14]. Recently, the edge metric dimension becomes a very common topic in resolvability and a lot of families of graphs are studied. Koam and Ahmad [15] studied edge metric dimension of barycentric subdivision of Cayley graph. e convex polytope graph was discussed by Zhang and Gao in [16] and Ahsan et al. in [17]. Yang et al. discussed some chemical structures related to wheel graphs in [18]. Raza and Bataineh did comparative analysis between metric and edge metric dimension in [19]. Moreover, some interesting study of edge metric dimension can be found in [20,21], where Okamoto et al. studied the local metric dimension and Yero briefly discussed the definition of metric and its related concepts. Mixed metric dimension is another type of dimension which satisfies the conditions of metric and edge metric dimensions simultaneously. Mixed metric dimension of different families of graphs is studied and gives their exact values, such as Raza et al. studied the mixed metric dimension different rotationally symmetric graphs and gave their exact values [22], Raza and Ji computed the mixed metric dimension of the generalize Petersen graph P(n, 2) [23], and results on mixed metric dimension of some pathrelated graphs are discussed by Raza et al. in [24].
In general, the edge metric dimension of a graph is NPhard [3]. ere is no general relation between metric and edge metric dimension of graphs but, in [3], Kelenc et al. inquired about the families of graphs which have dim(G) � dim e (G), dim(G) < dim e (G), and dim(G) > dim e (G). In this paper, we find the edge metric dimension of different families of Möbius ladder networks and give a comparison between the metric and edge metric dimension of these families, in the response of this question.

Definition 1.
e minimum number of edges h between two vertices a 1 , a 2 of a cycle (sub) graph is called as h-size gap between a 1 , a 2 .

Edge Metric Dimension of Möbius Ladder Network
Möbius ladder ML ψ is built by a grid of ψ × 1, and this grid is twisted at 180 ∘ ; now, paste the extreme most left and right paths of vertices as seen in Figure 1. It contains ψ-horizontal cycles of order four. e metric dimension of ML ψ is three [9], and in our first result, we prove that the edge metric dimension of ML ψ is four.

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Mathematical Problems in Engineering From the above given representation of all edges with respect to edge resolving set Q e , no two edges have the same representation, and it proved that dim e (ML ψ ) ≤ 4. Now, on contrary, dim e (ML ψ ) � 3 implies that the cardinality of edge metric generator Q e ′ is three, following is some discussion for this contradiction: Case 1: if the first two vertices with zero-size gap and last two vertices with any arbitrary size of gap are selected in the edge metric generator Q e ′ � ρ 1 , ρ 2 , ρ ω with 3 ≤ ω ≤ ψ, then it implies the same representations in the edges d(ρ 1 ρ ψ+1 |Q e ′ ) � d(ρ 1 ρ 2ψ |Q e ′ ), and it is concluded that we cannot take these types of vertices in the edge metric generator with cardinality three. Case 2: if all three vertices with any arbitrary size of gap are selected and Q e ′ � ρ ω , ρ j , ρ k with 1 ≤ ω, j, k ≤ ψ, then it implies the same representations in the edges which are d(ρ 1 ρ ψ+1 |Q e ′ ) � d(ρ 1 ρ 2ψ |Q e ′ ), and it is also concluded that we cannot take these types of vertices in the edge metric generator with cardinality three. Case 3: now, the vertices with index ψ + 2 ≤ ω, j, k ≤ 2ψ with any of the gap-size are chosen and Q e ′ � ρ ω , ρ j , ρ k , then it implies the same representations in the edges which are d(ρ 1 ρ 2 |Q e ′ ) � d(ρ 1 ρ ψ+1 |Q e ′ ), and again, we conclude that we cannot take these types of vertices in the edge metric generator with cardinality three. Case 4: now, the vertices ρ ω , ρ j , ρ k ∈ V(ML ψ ) are chosen with any of the size of gap, then there exist an edge from lower horizontal edges and another edge from upper horizontal edges having the same representations to each other with respect to the selected edge metric generator, i.e., d(ρ p ρ p+1 |Q e ′ ) � d(ρ ψ+p ρ ψ+p+1 |Q e ′ ) where 1 ≤ p ≤ ψ − 1, and finally, we concluded that with any gap-size in the edge metric generator with cardinality three is not possible, which implies the contradiction that dim e (ML ψ ) ≠ 3. Moreover, this proves the double inequality which is dim e ML ψ � 4.

Edge Metric Dimension of Hexagonal Möbius Ladder Network
Hexagonal Möbius ladder HML ψ is built in [25], it can be constructed by dividing each horizontal edge of a square grid by inserting a new vertex, it becomes a grid of ψ × 1 with each cycle having order six, and now, twist this grid at 180 ∘ and paste the extreme most left and right paths of vertices as shown in Figure 2. is graph contains ψ-horizontal cycles of order six. e metric dimension of the hexagonal Möbius ladder network is three [25]. In this section, we proved that edge metric dimension is also three for the hexagonal Möbius ladder network.

Edge Metric Dimension of Triangular
Ladder Network e ladder network can be built by the cross product of two path graphs L ψ � P ψ × P 2 . Triangular ladder TL ψ is built by inserting new edges of L ψ vertices ρ ω ρ ω+1 where ω is even indices.
is graph contains 2ψ − 2-cycles of order three shown in Figure 3. Following is the edge metric generator of this network.

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Proof. Let Q e � ρ 1 , ρ 2 , ρ 2ψ−2 , ρ 2ψ−1 be the edge resolving set. To show that the dim e (TL ψ ) � 4, we will prove first for dim e (TL ψ ) ≤ 4, following are the representations of all edges with respect to edge resolving set: Above given representations of all edges with respect to edge resolving set Q e are unique, and it is proved that dim e (TL ψ ) ≤ 4. Now, for dim e (TL ψ ) ≥ 4, choose on contrary which implies that dim e (TL ψ ) � 3. Analogously, the cardinality of edge metric generator Q e ′ is three, following is some discussion for this contradiction: Case 1: if all vertices with any arbitrary size of gap are selected and Q e ′ � ρ 1 , ρ ω , ρ j with 3 ≤ ω, j(odd) ≤ 2ψ − 1, then it implies the same representations in the edges which are d(ρ 2 ρ 3 |Q e ′ ) � d(ρ 3 ρ 4 |Q e ′ ), and it is concluded that we cannot take vertices like this in the edge metric generator with cardinality three. Case 2: if all vertices with any arbitrary size of gap are selected and Q e ′ � ρ ω , ρ j , ρ k with 1 ≤ ω, j, k(odd) ≤ 2ψ − 1, then it implies the same representations in the edges which are either d(ρ 1 ρ 3 |Q e ′ ) � d(ρ 2 ρ 3 |Q e ′ ) or d(ρ p ρ p+1 |Q e ′ ) � d(ρ q ρ q+2 |Q e ′ ) where 2 ≤ p, q(even) ≤ 2ψ, and it is also concluded that we cannot take vertices in the edge metric generator with cardinality three. Case 3: if all vertices with any arbitrary size of gap are selected and Q e ′ � ρ ω , ρ j , ρ k with 2 ≤ ω, j, k(even) ≤ 2ψ, then it implies the same representations in the edges which are d(ρ 1 ρ 2 |Q e ′ ) � d(ρ 2 ρ 3 |Q e ′ ), and it is also concluded that we cannot take vertices in the edge metric generator with cardinality three. Case 4: now, the vertices ρ ω , ρ j , ρ k ∈ V(TL ψ ) without choosing the size of gap, then there exist an edge from joining edges and another one edge from upper horizontal edges having the same representation to each other with respect to the decided edge metric generator, i.e., d(ρ p ρ p+1 |Q e ′ ) � d(ρ q ρ q+2 |Q e ′ ) where 2 ≤ p, q(even) ≤ 2ψ, and finally, we concluded that we cannot take any type of vertices with any gap-size in the edge metric generator with cardinality three, which implies the contradiction that dim e (TL ψ ) ≠ 3, Moreover, this proves the double inequality which is dim e TL ψ � 4.  Figure 3 shows a triangular ladder network; now, twist this network at 180 ∘ and paste the extreme most left and right paths of vertices, and it will come up to a new type of graph named as triangular Möbius ladder graph TML ψ . is graph contains 2ψ − 2-horizontal cycles of order three, which can be seen in Figure 4. Following results are the edge metric dimension of the triangular Möbius ladder network. Mathematical Problems in Engineering