On Constant Metric Dimension of Some Generalized Convex Polytopes

<jats:p>Metric dimension is the extraction of the affine dimension (obtained from Euclidean space <jats:inline-formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" id="M1">
                        <msup>
                           <mrow>
                              <mi>E</mi>
                           </mrow>
                           <mrow>
                              <mi>d</mi>
                           </mrow>
                        </msup>
                     </math>
                  </jats:inline-formula>) to the arbitrary metric space. A family <jats:inline-formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" id="M2">
                        <mi mathvariant="normal">ℱ</mi>
                        <mo>=</mo>
                        <mfenced open="(" close=")" separators="|">
                           <mrow>
                              <msub>
                                 <mrow>
                                    <mi>G</mi>
                                 </mrow>
                                 <mrow>
                                    <mi>n</mi>
                                 </mrow>
                              </msub>
                           </mrow>
                        </mfenced>
                     </math>
                  </jats:inline-formula> of connected graphs with <jats:inline-formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" id="M3">
                        <mi>n</mi>
                        <mo>≥</mo>
                        <mn>3</mn>
                     </math>
                  </jats:inline-formula> is a family of constant metric dimension if <jats:inline-formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" id="M4">
                        <mi mathvariant="normal">dim</mi>
                        <mfenced open="(" close=")" separators="|">
                           <mrow>
                              <mi>G</mi>
                           </mrow>
                        </mfenced>
                        <mo>=</mo>
                        <mi>k</mi>
                     </math>
                  </jats:inline-formula> (some constant) for all graphs in the family. Family <jats:inline-formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" id="M5">
                        <mi mathvariant="normal">ℱ</mi>
                     </math>
                  </jats:inline-formula> has bounded metric dimension if <jats:inline-formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" id="M6">
                        <mi mathvariant="normal">dim</mi>
                        <mfenced open="(" close=")" separators="|">
                           <mrow>
                              <msub>
                                 <mrow>
                                    <mi>G</mi>
                                 </mrow>
                                 <mrow>
                                    <mi>n</mi>
                                 </mrow>
                              </msub>
                           </mrow>
                        </mfenced>
                        <mo>≤</mo>
                        <mi>M</mi>
                     </math>
                  </jats:inline-formula>, for all graphs in <jats:inline-formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" id="M7">
                        <mi mathvariant="normal">ℱ</mi>
                     </math>
                  </jats:inline-formula>. Metric dimension is used to locate the position in the Global Positioning System (GPS), optimization, network theory, and image processing. It is also used for the location of hospitals and other places in big cities to trace these places. In this paper, we analyzed the features and metric dimension of generalized convex polytopes and showed that this family belongs to the family of bounded metric dimension.</jats:p>


Introduction
Let G ∈ F be a finite, simple, and undirected connected graph with vertex set V � V(G) � v 1 , v 2 , . . . , v n and edge set E � E(G). e distance between two vertices is denoted by d(v s , v j ) � d sj where d sj is the length of the shortest path between these vertices in G. Moreover, the distance d sj � d js because all graphs are undirected. An ordered subset W � w 1 , w 2 , . . . , w k of V is called a resolving set or locating set for G if for any two distinct vertices v s and v j , their codes are distinct with respect to Z, where code(v s ) � (d(v s , z 1 ), d(v s , z 2 ), . . . , d(v s , z k )) ∈ W k is a vector [1]. min : |W|: W is a resolving set of G � dim(G) � β(G) is called the metric dimension or locating number of G, and such a resolving set Z is called a basis set for G. To investigate Z is a basis set for G, it suffices to show that, for all different vertices x, y ∈ V∖W, their codes are also different because for any w j ∈ W, 1 ≤ j ≤ k, the jth component of the code is zero, while all other components are nonzero. For more details about β(G) and resolving sets, one can read [1][2][3][4].
Lemma 1 (see [3]). For a connected graph G with resolving e join of two graphs G and H represented as W n � C n + K 1 is a wheel graph of order n + 1 for n ≥ 3. f n � P n + K 1 is a fan graph obtained from the amalgamation of the path on n vertices with a single vertex graph K n . Jahangir or gear graph J 2n is obtained from the wheel graph W 2n by deleting n-cycle edges alternatively; see in [4]. e following results appear in [5][6][7] for the graphs defined above. Theorem 1. For wheel graph W n , fan graph f n , and Jahangir graph J 2n , we have the following: (iii) β(J 2n ) � ⌊2n/3⌋, for every n ≥ 4 All the above three families of graphs are planar, and their metric dimension depends on the number of vertices in the graph, which shows that the metric dimension of these graphs is unbounded [8,9]. Khuller et al. [10] clarified the properties of those graphs whose metric dimension is two.
Theorem 2 (see [10]). Let β(G) � 2 � |W|, where W � x, y ⊂ V(G); then, the following holds: (i) ere is a unique shortest path P between x and y (ii) deg(x) ≤ 3 or deg(y) ≤ 3 (iii) For every other vertex z except x and y on P, deg(z) ≤ 5 Definition 1 (see [11]). A set K ⊂ R d is said to be convex if the line segment xy: λx Definition 2 (see [11]). e smallest convex set containing K (the intersection of the family of all convex sets that contain Definition 3 (see [11]). A convex polytope is a bounded convex linear combination of convex sets.
ere are some families of graphs with constant metric dimension (see [2]); these families are generated by convex polytopes. e problem of finding β(G) is NP-complete (see [2]).
Theorem 3 (see [12]). Let S p n be a convex polytope with p-pendent vertices; then, dim(S p n ) � 3 for all n ≥ 6.
Theorem 4 (see [12]). e metric dimension of convex polytope T p n with p-pendent edges is 3 for every n ≥ 6.
Theorem 5 (see [12] For more details about the metric dimension of certain families of graphs, see [13,14]. Here, we will investigate generalized convex polytopes with pendent edges for their metric dimensions.

Main Results
is section is devoted to the main results which we proved for the newly introduced generalized convex polytopes. e convex polytopes S n , T n , and U n were examined by Muhammad et al. for their metric dimensions in [2] and proved that these families belong to the family of constant metric dimension.
Generalized convex polytope S n,m is the generalization of S n , with one n-sided and infinite face each, 3-sided faces being 2n, and 4-sided faces being n(m − 2), so the total number of faces is nm + 2. e convex polytope S p n,m is obtained from the generalized convex polytope graph by attached p-pendent vertices at the outer cycle of S n,m , shown in Figure 1. e generalized convex polytope S p n.m with p-pendents is a graph consisting of m cycles, with vertex and edge sets In the set of edges, indices are taken as modulo n and m.
In [2], it was shown that β(S n ) � 3, for n ≥ 6. In the result below, we proved that the metric dimension for the generalized convex polytope of S n is still 3, which implies that S n , Proof. Validating the mentioned theorem with the help of double inequalities, two cases are present: Case (i): for n is even. Let n � 2α ′ where α ′ ≥ 3 is an integer. As |N 2 (x)| ≥ 6 for all x ∈ S n,m , it is guaranteed by [15] that β(G) ≥ 3. Consider Z � X 1 1 , X 1 2 , X 1 l+1 to be an ordered subset of V(S p n,m ); to show that Z is a basis set for G, codes of the elements of V(S p n,m )∖Z with respect to Z are given in the following scheme: Codes for the vertices X m s for 1 ≤ s ≤ n and m ≥ 3 are given in the following: It proves that β(G) ≤ 3 implies that the metric dimension of G � S p n,m is 3. Case (ii): for n is an odd integer.
Let n � 2α ′ + 1, where α ′ ≥ 3, and by [15], β(G) ≥ 3; for reaching the conclusion, it remains to show that β(G) ≤ 3. Let Z � X 1 1 , X 1 2 , X 1 l+1 be an ordered subset of S p n,m ; the formulation for the representation of nodes for V(S p n,m )∖Z with respect to Z is given in the following: e representation of the vertices X m s , 1 ≤ s ≤ n and m ≥ 3, is as follows: It shows that, for any two distinct vertices x, y ∈ S p n,m for odd n ≥ 7, r(x|Z) ≠ r(y|Z) implying that β(G) ≤ 3; this completes the proof. e general form of T n is denoted by T n,m known as the generalized convex polytope (for short, GCP); this graph consists of one each n-sided and infinite face, respectively, and the number of 3-sided faces is 4n and 4sided faces is n(m − 3). e GCP graph T p n,m is a graph with p-pendent edges. Vertex and edge sets for G � T p n,m are given in the following: In the set of edges, indices are taken as modulo n and m. In Figure 2, the graph G � T p n,m is shown. e result given below shows that T p n,m belongs to the family of constant metric dimension.  [15]. To complete the proof, it suffices to show that any ordered subset of the vertices of this graph is a resolving set. Case (i): for n is an even integer. Let n � 2α ′ with α ′ ≥ 3; consider an ordered subset Z � X 1 1 , X 1 2 , X 1 l+1 of vertices of T p n,m . e representation of vertices of V(T p n,m )∖Z with respect to Z is formulated as follows: Codes of the pendent vertices are given as follows: r X m s |Z � (1, 1, 1) + r X m−1 s |Z .
From the above formulation, it is obvious that no two distinct vertices of the GCP with pendents p have the same code with respect to Z, which implies that β(T p n,m ) � 3.
Case (ii): for n is an odd integer. Let n � 2α ′ + 1 for α ′ ≥ 3; suppose an ordered subset Z � x 1 1 , X 1 2 , X 1 l+1 of vertices V(T p n,m ); to show that Z is a basis set for T p n,m , the formulation codes are given as follows: