Research Article

1.   Introduction

It has been observed that the graph of an 𝑘-gonal (where 𝑘∈N and 𝑘≥3) prism exhibits a distinct pattern in terms of its vertices and edges. Specifically, it has been determined that an 𝑘-gonal prism possesses 2𝑘 vertices and 3𝑘 edges. This finding provides valuable insight into the structural properties of 𝑘-gonal prisms and contributes to our understanding of their geometric characteristics. The graphs in question exhibit regularity and possess a cubic structure. Prism graphs possess the property of vertex-transitivity due to the presence of symmetries that map each vertex to every other vertex. As polyhedral graphs, they exhibit the property of being 3-vertex-connected planar graphs. It has been observed that every prism graph, which is a specific type of graph formed by connecting two copies of a cycle graph with corresponding vertices, possesses a Hamiltonian cycle. A Hamiltonian cycle is a cycle that visits each vertex exactly once. Researchers have extensively studied this property and proven its validity for all prism graphs [1].

Throughout this article, all graphs under consideration are connected, non-trivial, undirected, and simple. Let Γ_𝑟 =(𝑉,𝐸) be a graph with 𝑉(Γ_𝑟) and 𝐸(Γ_𝑟) as its vertex and edge set respectively. The shortest length path between two distinct vertices 𝑦₁ and 𝑦₂ in 𝑉(Γ_𝑟), is referred to as the distance (𝑑(𝑦₁,𝑦₂)) between 𝑦₁ and 𝑦₂ in Γ_𝑟. The number of distinct edges incident on a vertex 𝑦 in Γ_𝑟 is known as the degree of 𝑣 (denoted by 𝑑_𝑦). Two vertices 𝑦₁ and 𝑦₂ in Γ_𝑟 are said to resolved by a vertex 𝑦, if 𝑑(𝑦,𝑦₁)≠ 𝑑(𝑦,𝑦₂) in Γ_𝑟. Then, a subset 𝑈⊆𝑉(Γ_𝑟) with this property, i.e., every pair of unequal vertices in Γ_𝑟 can be resolved by at least one member of 𝑈, is said to be a resolving set (RS) for Γ_𝑟. The smallest cardinality set 𝑈 with resolving characteristic is called the metric basis (MB) for Γ_𝑟, and the MB set cardinality is the metric dimension (MD) for Γ_𝑟, represented by 𝑑𝑖𝑚(Γ_𝑟) [2, 3].

For a subset 𝑈={𝑦₁,𝑦₂,𝑦₃,...,𝑦_𝑠} of distinct ordered vertices in 𝑉(Γ_𝑟), the unique 𝑠-length tuple code for each 𝑞∈𝑉(Γ_𝑟) is given as follows ζ(𝑞|𝑈)=(𝑑(𝑞,𝑦₁),𝑑(𝑞,𝑦₂),𝑑(𝑞,𝑦₃),...,𝑑(𝑞,𝑦_𝑠)). Using this fact, the subset 𝑈 is a RS for Γ_𝑟, if ζ(𝑦₁|𝑈)≠ ζ(𝑦₂|𝑈), for every pair of different vertices 𝑦₁,𝑦₂ ∈𝑉(Γ_𝑟). Next, a subset 𝑈 in 𝑉(Γ_𝑟) with distinct vertices is said to be a resolving independent set for Γ_𝑟, if it is (i) independent set as well as (ii) a RS in Γ_𝑟. A proper subset of a RS is not necessarily a RS, while a superset of every RS is always aRS [4].

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Figure 1: Graph Γ^*

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To understand the concept of RS and MD, let us consider a graph Γ^* on 5 vertices and 7 edges, as shown in Fig. 1. To find the metric dimension of Γ^*, we suppose that 𝑈₁ ={𝑣₁,𝑣₄} (red color vertices in Γ^*). Next, metric codes for each vertex in Γ^* with respect to 𝑈₁ are as follows: ζ(𝑣₁|𝑈₁)=(0,1), ζ(𝑣₂|𝑈₁)=(1,2), ζ(𝑣₃|𝑈₁)=(2,2), ζ(𝑣₄|𝑈₁)=(1,0), and ζ(𝑣₅|𝑈₁)=(1,1). From this, we find that the metric codes for all the vertices in Γ^* corresponding to the set 𝑈₁ are unique, and so we say that 𝑈₁ is a resolving set for Γ^*. Also, 𝑈₁ is the minimum resolving set for Γ^*, as the cardinality of 𝑈₁ is 2 [5]. Hence, we have concluded that 𝑑𝑖𝑚(Γ^*)=2.

Slater [3] and Harary & Melter [2] independently introduced the concept of MD of a graph in 1970s. There is a wide range of literature available on MD that addresses both theoretical and practical aspects. The MD has appeared in various areas including combinatorial optimization, sonar, pharmaceutical chemistry, robot navigation, graph isomorphism testing, and many more see [6, 7, 8, 9, 10] and references therein.

The computation of MD for distinct graph families, is always a challenging task because deciding and selecting of a landmark (resolving) vertices in minimum numbers is not that easy due to the complexity and scalability of the considered graph network. Further, the notion of MD was extended as well as investigated by eminent researchers from time to time and named them as the variants of MD [11]. Several authors have studied MD and its related variants for several distinct graphs as well as for various other graph-theoretic aspects, for instance prism graph, path graph, complete graph, cycle graph, cycle graph with chords, antiprism graph, several ladder graphs (pentagonal, heptagonal, etc), convex polytope graph, wheel graph, tadpole graph, kayak paddle graph, and numerous planar and chemical graphs [4, 6, 7, 9, 10, 12, 13]. Even though after investigating these notions for the large number of graph families, there are still many families for which these notions are not investigated to date.
In this paper, two planar graph families, viz., L_𝑛 and M_𝑛 has been constructed, which are obtained by taking 𝑚-copies (𝑞= 𝑚) of the prism graph on the two known convex polytope graphs N_𝑛 [9] and 𝑈_𝑛 [7], respectively. For these so obtained graphs, we investigate their minimal MB sets and finally obtain their MD. To carry out these results, we need the following result:

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Figure 2: The Graph L_𝑛

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The following paper is structured as follows: In Section 2, we consider an infinite family of convex polytope L_𝑛 and find its minimum RS with its respective MD. In Section 3, we consider an infinite family of convex polytope M_𝑛 and find its minimum RS with its respective MD. Finally, the conclusion and future scope of the manuscript is presented.

2.   Minimum Vertex Resolving Number of L_𝑛

In this section, we investigate some of the basic properties and the MD of a planar graph L_𝑛. The graph L_𝑛 consists of 𝑛(𝑞+3) vertices and 𝑛(2𝑞+5) edges (see Fig.2). The sets containing vertices and edges for planar graph L_𝑛 are denoted by 𝑉(L_𝑛) and 𝐸(L_𝑛) respectively, where 𝐸(L_𝑛)={𝑗_α 𝑗_α+1 ,𝑗_α 𝑘_α ,𝑘_α 𝑗_α+1 ,𝑘_α 𝑙_α ,𝑙_α 𝑚_α ¹,𝑚_α ¹𝑙_α+1 ,𝑚_α ^𝑠𝑚_α+1 ^𝑠 : 1 ≤α≤𝑛,1 ≤𝑠≤𝑞}∪{𝑚_α ^𝑠𝑚_α ^𝑠+1 : 1 ≤α≤𝑛,1 ≤𝑠≤𝑞-1} and 𝑉(L_𝑛)={𝑗_α ,𝑘_α ,𝑙_α ,𝑚_α ^𝑠 : 1 ≤α≤𝑛,1 ≤𝑠≤𝑞}.

We call vertices {𝑗_α : 1 ≤α≤𝑛}, as 𝑗-cycle vertices in L_𝑛, the vertices {𝑘_α ,𝑙_α : 1 ≤α≤𝑛}, as inner vertices in L_𝑛, and the vertices {𝑚_α ^𝑠 : 1 ≤α≤𝑛,1 ≤𝑠≤𝑞}, as outer vertices in L_𝑛. In the following result, we investigate the MD of L_𝑛.

Proof. Now, the following cases, which depend on the natural 𝑛, can be employed to investigate this result.

Case(I) 𝑛≡0(𝑚𝑜𝑑 2).
We set 𝑛=2𝑦; 𝑦∈Z⁺ and 𝑦≥3. Let 𝑈={𝑗₂,𝑗_𝑦+1,𝑗_𝑛}⊂𝑉(L_𝑛). Next, each vertex of L_𝑛 has given metric coordinate corresponding to the taken set 𝑈.
For vertices over 𝑗-cycle, i.e., {𝑗_α : 1 ≤α≤𝑛}, the metric co-ordinates are

For the vertices {𝑘_α : 1 ≤α≤𝑛}, the metric co-ordinates are

For the vertices {𝑙_α : 1 ≤α≤𝑛}, the metric co-ordinates are ζ(𝑙_α |𝑈)= ζ(𝑘_α |𝑈)+(1,1,1) for 1 ≤α≤𝑛. Next, for the vertices {𝑚_α ¹ : 1 ≤α≤𝑛}, the metric co-ordinates are

Finally, for the vertices {𝑚_α ^𝑠 : 1 ≤α≤𝑛,2 ≤𝑠≤𝑞}, the co-ordinates are ζ(𝑚_α ^𝑠|𝑈)=ζ(𝑚_α ¹|𝑈)+(𝑠-1,𝑠-1,𝑠-1) for 1 ≤α≤𝑛. Next, these codes for all vertices in L_𝑛 are unique and distinct from one and an other in at least one co-ordinate, which results in 𝑑𝑖𝑚(L_𝑛)≤3.

Now, for reverse inequality i.e., 𝑑𝑖𝑚(L_𝑛)≥3, we show that no set 𝑈 with |𝑈|=2, form a RS for L_𝑛. Assuming 𝑑𝑖𝑚(L_𝑛)=2 (on contrary). Using proposition 1, we have following to discuss for RS 𝑈 with |𝑈|= 2 in L_𝑛:

1. Let 𝑈= {𝑘₁,𝑘_𝑔}, 𝑘_𝑔 (2 ≤ 𝑔≤ 𝑦+1), then ζ(𝑗_𝑛|𝑈) = ζ(𝑘_𝑛|𝑈), for 2 ≤ 𝑔≤ 𝑦-1, ζ(𝑙₂|𝑈) = ζ(𝑘_𝑛-1|𝑈), when 𝑔= 𝑦, and ζ(𝑗₂|𝑈) = ζ(𝑗₁|𝑈), when 𝑔= 𝑦+ 1, a contradiction.

2. Let 𝑈= {𝑙₁,𝑙_𝑔}, 𝑙_𝑔 (2 ≤ 𝑔≤ 𝑦+1), then ζ(𝑗_𝑛|𝑈) = ζ(𝑘_𝑛|𝑈), for 2 ≤ 𝑔≤ 𝑦-1, ζ(𝑝₃|𝑈) = ζ(𝑚₂²|𝑈), when 𝑔= 𝑦, and ζ(𝑗₂|𝑈) = ζ(𝑗₁|𝑈), when 𝑔= 𝑦+ 1, a contradiction.

3. Let 𝑈= {𝑚₁^𝑞,𝑚_𝑔^𝑞}, 𝑚_𝑔^𝑞 (2 ≤ 𝑔≤ 𝑦+1), then ζ(𝑚₁^𝑞-1|𝑈)= ζ(𝑚_𝑛^𝑞|𝑈), for 2 ≤𝑔≤𝑦, and ζ(𝑚₂^𝑞|𝑈)= ζ(𝑚_𝑛^𝑞|𝑈), when 𝑔= 𝑦+1, a contradiction.

4. Let 𝑈= {𝑘₁,𝑙_𝑔}, 𝑙_𝑔 (1 ≤ 𝑔≤ 𝑦+1), then ζ(𝑗_𝑛|𝑈) = ζ(𝑘_𝑛|𝑈), for 1 ≤ 𝑔≤ 𝑦-1, ζ(𝑚₁¹|𝑈) = ζ(𝑝₃|𝑈), when 𝑔= 𝑦, and ζ(𝑗₂|𝑈) = ζ(𝑗₁|𝑈), when 𝑔= 𝑦+ 1, a contradiction.

5. Let 𝑈= {𝑘₁,𝑚_𝑔^𝑞}, 𝑚_𝑔^𝑞 (1 ≤𝑔≤𝑦+1), then ζ(𝑗₁|𝑈) = ζ(𝑗₂|𝑈), for 𝑔= 1, ζ(𝑚_𝑛¹|𝑈) = ζ(𝑘₂|𝑈), when 2 ≤𝑔≤𝑦, and ζ(𝑚_𝑦+1¹|𝑈)= ζ(𝑙_𝑦+2|𝑈), when 𝑔=𝑦+1, a contradiction.

6. Let 𝑈= {𝑙₁,𝑚_𝑔^𝑞}, 𝑚_𝑔^𝑞 (1 ≤𝑔≤𝑦+1), then ζ(𝑗₁|𝑈) = ζ(𝑗₂|𝑈), for 𝑔= 1, ζ(𝑚_𝑛-1¹|𝑈) = ζ(𝑗₂|𝑈), when 2 ≤ 𝑔≤ 𝑦-1, ζ(𝑚_𝑦¹|𝑈) = ζ(𝑚_𝑦-1²|𝑈), when 𝑔= 𝑦, and ζ(𝑚_𝑦+1¹|𝑈) = ζ(𝑚_𝑦+2²|𝑈), when 𝑔=𝑦+1, a contradiction.

Thus, from this we have 𝑑𝑖𝑚(L_𝑛)≥3, implying that 𝑑𝑖𝑚(L_𝑛)=3, ∀𝑛≥6.

Case(II) 𝑛≡1(𝑚𝑜𝑑 2).

We set 𝑛= 2𝑦+1; 𝑦∈Z⁺ and 𝑦≥3. Let 𝑈={𝑗₂,𝑗_𝑦+1,𝑗_𝑛}⊂𝑉(L_𝑛). Next, each vertex of L_𝑛 has given metric coordinate corresponding to the taken set 𝑈.

For vertices over 𝑗-cycle, i.e., {𝑗_α : 1 ≤α≤𝑛}, the metric co-ordinates are

For the vertices {𝑘_α : 1 ≤α≤𝑛}, the metric co-ordinates are

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Figure 3: The Graph M_𝑛

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Forthe vertices {𝑙_α : 1 ≤α≤𝑛}, the metric co-ordinates are ζ(𝑙_α |𝑈)=ζ(𝑘_α |𝑈)+(1,1,1) for 1 ≤α≤𝑛. Next, for the vertices {𝑚_α ¹ : 1 ≤α≤𝑛}, the co-ordinates are

Finally, for the vertices {𝑚_α ^𝑠 : 1 ≤α≤𝑛,2 ≤𝑠≤𝑞}, the metric co-ordinates are ζ(𝑚_α ^𝑠|𝑈)=ζ(𝑚_α ¹|𝑈)+(𝑠-1,𝑠-1,𝑠-1) for 1 ≤α≤𝑛. Next, these codes for all vertices in L_𝑛 are unique and distinct from one and an other in at least one co-ordinate, which results in 𝑑𝑖𝑚(L_𝑛)≤3. Assuming that 𝑑𝑖𝑚(L_𝑛)= 2, then as in Case (I), we have the same contradictions. Therefore, we have 𝑑𝑖𝑚(L_𝑛)= 3 as well in this case, which proofs the theorem. □

3.   Minimum Vertex Resolving Number of M_𝑛

In this section, we investigate some of the basic properties and the MD of a planar graph M_𝑛 (see Fig. 3). The graph M_𝑛 consists of 𝑛(𝑞+4) vertices and 𝑛(2𝑞+6) edges. The sets containing vertices and edges for planar graph M_𝑛 are denoted by 𝑉(M_𝑛) and 𝐸(M_𝑛) respectively, where 𝐸(M_𝑛)={𝑗_α 𝑗_α+1 ,𝑗_α 𝑘_α ,𝑘_α 𝑘_α+1 ,𝑘_α 𝑙_α ,𝑙_α 𝑚_α ,𝑚_α 𝑙_α+1 ,𝑚_α 𝑜_α ¹,𝑜_α ^𝑠𝑜_α+1 ^𝑠 : 1 ≤α≤𝑛,1 ≤𝑠≤ 𝑞}∪{𝑜_α ^𝑠𝑜_α ^𝑠+1 : 1 ≤α≤𝑛,1 ≤𝑠≤𝑞-1} and 𝑉(M_𝑛)={𝑗_α ,𝑘_α ,𝑙_α ,𝑚_α ,𝑜_α ^𝑠 : 1 ≤α≤𝑛,1 ≤𝑠≤𝑞}.
We call vertices {𝑗_α : 1 ≤α≤𝑛}, as 𝑗-cycle vertices in M_𝑛, the vertices {𝑘_α : 1 ≤α≤𝑛}, as 𝑘-cycle vertices in M_𝑛, the vertices {𝑙_α ,𝑚_α : 1 ≤α≤𝑛}, as 𝑙𝑚-cycle vertices in M_𝑛, and the vertices {𝑜_α ^𝑠 : 1 ≤α≤𝑛,1 ≤𝑠≤𝑞} as outer vertices in M_𝑛. In the following result, we investigate the MD of M_𝑛.

Proof. Now, the following cases, which depend on the natural 𝑛, can be employed to investigate this result.

Case(I) 𝑛≡0(𝑚𝑜𝑑 2).
We set 𝑛= 2𝑦; 𝑦∈ Z⁺ and 𝑦≥ 3. Let 𝑈={𝑗₂,𝑗_𝑦+1,𝑗_𝑛}⊂𝑉(M_𝑛). Next, each vertex of M_𝑛 has given metric coordinate corresponding to the taken set 𝑈.
For vertices over 𝑗-cycle, i.e., {𝑗_α : 1 ≤α≤𝑛}, the metric co-ordinates are

For the vertices {𝑘_α : 1 ≤α≤𝑛}, the metric co-ordinates are ζ(𝑘_α |𝑈)= ζ(𝑗_α |𝑈)+(1,1,1) for 1 ≤α≤𝑛. Next, for the vertices {𝑙_α : 1 ≤α≤𝑛}, the metric co-ordinates are ζ(𝑙_α |𝑈)= ζ(𝑘_α |𝑈)+(1,1,1) for 1 ≤α≤𝑛.
For the vertices {𝑚_α : 1 ≤α≤𝑛}, the metric co-ordinates are

Finally, for the vertices {𝑜_α ^𝑠 : 1 ≤α≤𝑛,1 ≤𝑠≤𝑞}, the metric co-ordinates are ζ(𝑜_α ^𝑠|𝑈)=ζ(𝑚_α |𝑈)+(𝑠,𝑠,𝑠) for 1 ≤α≤𝑛.

Next, these codes for all vertices in M_𝑛 are unique and distinct from one and an other in at least one co-ordinate, which results in 𝑑𝑖𝑚(M_𝑛)≤3. Now, for reverse inequality i.e., 𝑑𝑖𝑚(M_𝑛)≥3, we show that no set 𝑈 with |𝑈|= 2, form a RS for M_𝑛. Now, for reverse inequality i.e., 𝑑𝑖𝑚(M_𝑛)≥3, we show that no set 𝑈 with |𝑈|= 2, form a RS for M_𝑛. Assuming 𝑑𝑖𝑚(M_𝑛)=2 (on contrary). Using proposition 1, we have following to discuss for RS 𝑈 with |𝑈|= 2 in M_𝑛:

1. Let 𝑈= {𝑘₁,𝑘_𝑔}, 𝑘_𝑔 (2 ≤ 𝑔≤ 𝑦+1), then ζ(𝑗_𝑛|R) = ζ(𝑘_𝑛|R), for 2 ≤ 𝑔≤ 𝑦-1; ζ(𝑙₂|R) = ζ(𝑘_𝑛-1|R), when 𝑔= 𝑦; and ζ(𝑗₂|R) = ζ(𝑗₁|R), when 𝑔= 𝑦+1, a contradiction.

2. Let 𝑈= {𝑙₁,𝑙_𝑔}, 𝑙_𝑔 (2 ≤ 𝑔≤ 𝑦+1), then ζ(𝑗_𝑛|R) = ζ(𝑘_𝑛|R), for 2 ≤ 𝑔≤ 𝑦-1; ζ(𝑝₃|R) = ζ(𝑚₂²|R), when 𝑔= 𝑦; and ζ(𝑗₂|R) = ζ(𝑗₁|R), when 𝑔= 𝑦+ 1, a contradiction.

3. Let 𝑈= {𝑚₁^𝑞,𝑚_𝑔^𝑞}, 𝑚_𝑔^𝑞 (2 ≤ 𝑔≤ 𝑦+1), then ζ(𝑚₁^𝑞-1|R)= ζ(𝑚_𝑛^𝑞|R), for 2 ≤𝑔≤𝑦; and ζ(𝑚₂^𝑞|R)= ζ(𝑚_𝑛^𝑞|R), when 𝑔= 𝑦+1, a contradiction.

4. Let 𝑈= {𝑘₁,𝑙_𝑔}, 𝑙_𝑔 (1 ≤ 𝑔≤ 𝑦+ 1), then ζ(𝑗_𝑛|R) = ζ(𝑘_𝑛|R), for 1 ≤ 𝑔≤ 𝑦-1; ζ(𝑚₁¹|R) = ζ(𝑝₃|R), when 𝑔= 𝑦; and ζ(𝑗₂|R) = ζ(𝑗₁|R), when 𝑔= 𝑦+1, a contradiction.

5. Let 𝑈= {𝑘₁,𝑚_𝑔^𝑞}, 𝑚_𝑔^𝑞 (1 ≤ 𝑔≤ 𝑦+1), then ζ(𝑗₁|R)=ζ(𝑗₂|R), for 𝑔=1; ζ(𝑚_𝑛¹|R)= ζ(𝑘₂|R), when 2 ≤ 𝑔≤ 𝑦; and ζ(𝑚_𝑦+1¹|R) = ζ(𝑙_𝑦+2|R), when 𝑔=𝑦+1, a contradiction.

6. Let 𝑈= R = {𝑙₁,𝑚_𝑔^𝑞}, 𝑚_𝑔^𝑞 (1 ≤ 𝑔≤ 𝑦+1), then ζ(𝑗₁|R) = ζ(𝑗₂|R), for 𝑔= 1; ζ(𝑚_𝑛-1¹|R) = ζ(𝑗₂|R), when 2 ≤ 𝑔≤ 𝑦-1; ζ(𝑚_𝑦¹|R) = ζ(𝑚_𝑦-1²|R), when 𝑔= 𝑦; and ζ(𝑚_𝑦+1¹|R) = ζ(𝑚_𝑦+2²|R), when 𝑔= 𝑦+1, a contradiction.

Thus, from this we have 𝑑𝑖𝑚(M_𝑛)≥3, implying that 𝑑𝑖𝑚(M_𝑛)=3, ∀𝑛≥6.

Case(II) 𝑛≡1(𝑚𝑜𝑑 2).
We set 𝑛= 2𝑦+1, 𝑦∈ Z⁺ and 𝑦≥ 3. Let 𝑈={𝑗₂,𝑗_𝑦+1,𝑗_𝑛}⊂𝑉(M_𝑛). Next, each vertex of M_𝑛 has given metric coordinate corresponding to the taken set 𝑈.

For vertices over 𝑗-cycle, i.e., {𝑗_α : 1 ≤α≤𝑛}, the metric co-ordinates are

For the vertices {𝑘_α : 1 ≤α≤𝑛}, the metric co-ordinates are ζ(𝑘_α |𝑈)= ζ(𝑗_α |𝑈)+(1,1,1) for 1 ≤α≤𝑛. Next, for the vertices {𝑙_α : 1 ≤α≤𝑛}, the metric co-ordinates are ζ(𝑙_α |𝑈)= ζ(𝑘_α |𝑈)+(1,1,1) for 1 ≤α≤𝑛.

For the vertices {𝑚_α : 1 ≤α≤𝑛}, the metric co-ordinates are

Finally, for the vertices {𝑜_α ^𝑠 : 1 ≤α≤𝑛,1 ≤𝑠≤𝑞}, the metric co-ordinates are ζ(𝑜_α ^𝑠|𝑈)=ζ(𝑚_α |𝑈)+(𝑠,𝑠,𝑠) for 1 ≤α≤𝑛.

Next, these codes for all vertices in M_𝑛 are unique and distinct from one and an other in at least one co-ordinate, which results in 𝑑𝑖𝑚(M_𝑛)≤3. Assuming that 𝑑𝑖𝑚(M_𝑛)=2, then as in Case (I), we have the same contradictions. Therefore, we have 𝑑𝑖𝑚(M_𝑛)=3 as well in this case, which proves the theorem.
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4.   Conclusion and Discussion

Obtaining resolving set for novel planar structure plays an important role in interconnection networks for the transmission of the data. In this article, we proved that 𝑑𝑖𝑚(L_𝑛)=𝑑𝑖𝑚(M_𝑛)=3, where L_𝑛 and M_𝑛 are obtained by joining 𝑞-copies of the prism graph on N_𝑛 [9] and 𝑈_𝑛 [7], respectively. Further, we demonstrated for these two planar convex polytope graphs that the cardinality of respective resolving independent set is also three for them. Further, several other variations of MD were also introduced in last two decades, such as edge metric dimension, local metric dimension, strong metric dimension, mixed metric dimension, etc [11, 12, 14]. Therefore, in future we will try to investigate several other variants of MD for the planar graphs L_𝑛 and M_𝑛.

5.   Acknowledgements

We would like to express our sincere gratitude to the referees for their careful reading of this manuscript and for all of their insightful comments/criticism, which have resulted in a number of major improvements to this manuscript..

5.1.   Authorship contribution:

M. Gayathri is sole author of this article.

5.2.   Funding:

No funding was used in this study.

5.3.   Conflict of interest:

No conflicts of interest.

5.4.   Declaration:

This research has been conducted ethically, reporting of those involved in this article.

5.5.   Similarity Index:

I hereby confirm that there is no similarity index in abstract and conclusion while overall is less than 5% where individual source contribution is 2% or less than it.

5.6.   Data Availability

Data sharing is not applicable to this article as no data set were generated or analyzed during the current study.

References

Copyright

[© 2024 M. Gayathri] This is an Open Access article published in "Graduate Journal of Interdisciplinary Research, Reports & Reviews" (Grad.J.InteR³) ISSN(E): 2584-2919 by Vyom Hans Publications. It is published with a Creative Commons Attribution - CC-BY4.0 International License. This license permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

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