On Partition Dimension of Generalized Convex Polytopes

Let G be a graph having no loop or multiple edges


Introduction
Let us consider a connected graph G having fnite vertices and edges.Let θ and ϑ be the vertices in G, then d(θ, ϑ) is the distance between vertex θ and ϑ which is the shortest path between θ and ϑ.For the subset ℧ of G and vertex ϕ ∈ V(G).Te distance of vertex ϕ and set ℧ is defned as d(ϕ, ℧) � min d(ϕ, v)|v ∈℧  .Te order set ℧ � q 1 ,  q 2 , . . ., q l } of V(G) is referred to the l− vector r(ϕ|℧) � d(ϕ, q 1 ), d(ϕ, q 2 ), . . ., d(ϕ, q l )   as the representation of ϕ w.r.t ℧.Te set ℧ is said to be resolving set if ∀u ∈ G has diferent representations w.r.t ℧.Te minimum number of sets in resolving set is termed as metric dimension for G which is denoted by dim(G).Since 1975, the concept of metric dimension and metric bases was discussed in literature by diferent names.Te name of locating set was given by Slater [1].Melter and Harary introduce this concept by using term metric bases instead of locating set [2].In [3], Chartrand introduced this concept by the name of minimum resolving set.For more about resolving set and metric dimension, we refer [4][5][6][7][8][9].
As the partition of a set is the collection of its subset such that no two subsets overlap and the union of all such sets form the original set.Similarly, partition dimension is also concerned about partitioning of vertex set V(G) and resolvability.Te partition dimension is actually the generalization of metric dimension.For given l−ordered partition of vertices of G, where G is connected and simple is represented by c � c 1 , c 2 , . . ., c l  .Te representation for vertex ϕ ∈ V(G) is the vector r(ϕ|c) � (d (ϕ, c 1 ), d(ϕ, c 2 ), . . ., d(ϕ, c l )).Te partition c is the resolving partition if for all vertices in G this representation is unique w.r.t c. pd(G) is the smallest number of sets in resolving set c [10].Te problem of fnding the resolving set for a graph is NP-hard [11].As we know that, partition dimension for a graph is the generalization of metric dimension.Terefore, the problem of partition dimension is also NP-hard.
Graphs having n − 3 as partition dimension are discussed in [7].Graphs that are obtained by sum of path and cycle graph and its partition dimension are in [12,13], and also the bounds for partition dimension are provided.In [14,15], partition dimension of complete multipartite graphs is discussed, where strong partition dimension is discussed in [16,17].In [10], it is shown that the partition dimension of a graph G is bounded above by 1 more than its metric dimension.An upper bound for the partition dimension of a bipartite graph G is given in terms of the cardinalities of its partite sets, and it is shown that the bound is attained if and only if G is a complete bipartite graph.Graphs of order n having partition dimension 2, n, or n − 1 are characterized.In [18], the authors consider relationships between metric dimension, partition dimension, diameter, and other graph parameters.Tey constructed universal examples of graphs with given partition dimension and used these to provide bounds on various graph parameters based on metric and partition dimensions.In [19], authors studied the partition dimension of Cartesian product graphs.More precisely, they showed that for all pairs of connected graphs G and In [15], the authors studied the partition dimension of circulant graphs, which are Cayley graphs of cyclic groups.In [20], the authors found bounds for the cardinality of vertices in some wheel-related graphs, namely, gear graph, helm, sun fower, and friendship graph with given partition dimension k.In [21], the authors calculated the partition dimension of two (4, 6)−fullerene graphs.Tey also gave conjectures on the partition dimension of two (3, 6)−fullerene graphs.In [22], the authors obtained several tight bounds on the partition dimension of trees.In [23], the authors studied partition dimension of some families of convex polytopes with pendant edge and proved that these graphs have bounded partition dimension.In [24], sharp bounds for the fault tolerant partition dimension of certain wellknown families of convex polytopes are studied.Furthermore, it was studied that graphs having fault tolerant partition dimension are bounded below by 4. In [25], the authors considered the upper bound for the partition dimension of the generalized Petersen graph in terms of the cardinalities of its partite sets.In [17], the authors determined the partition dimension and strong metric dimension of a chain cycle constructed by even cycles and a chain cycle constructed by odd cycles [26] that mainly deal with metric dimension and partition dimension of tessellation of plane by boron nanosheets.It has been highlighted that there is a discrepancy between the mentioned parameters of the boron nanosheets.Moreover, some induced subgraphs of the stated sheets have been considered for the study of their metric dimension.
Tere are various applications of resolving partition in various felds and can be found in robot navigation, network discovery, network verifcation, in representing chemical compounds, strategies for the master mind games, Djokovic-Winkler relation, image processing and pattern recognition, and hierarchical data structure; for more applications of the desired study, we refer [2,[5][6][7][8][32][33][34].
In the study of the partition dimension for graph, the following theorems are very helpful.
Theorem 1 (see [13]).Let c be the resolving partition of then θ and ϑ be from diferent classes of c.
Theorem 2 (see [13]).Let G be a simple and connected graph, then If there is a constant β ≥ 1 having the property that pd(G) ≤ β, then partition dimension of K is bounded otherwise unbounded.Investigation of partition dimension of graphs is hard for some one, but one can easily compute bounds for the partition dimension in general family of graphs.From the research work given in [13], where the authors presented the graphs and the results in very organized way and found the upper bounds.In this work, we obtained upper bounds for various convex polytopes in their generalized form by adding prisms up to infnity.Te generalized form of the polytopes is studied and denoted by E n , G n , and S n .We found that partition dimension for the considered polytopes cannot be greater than 4. For lower bound of partition dimension, we present a consequence of Proposition 2.1 in the article [10], and we have that for a connected graph G, pd(G) ≥ 2 and equality holds for path graph of order n.

New Results
In this section, we investigated E n , G n , and S n in their generalized forms for partition dimension.We observed that partition dimension of these graphs is bounded by four, while generalization is made in terms of adding cycles that are extended into infnite numbers and can be seen in Figures 1-3.

Generalized
Convex Polytope E n .Te convex polytope E n is composed of two convex polytopes, antiprism A n and T n [13].Te generalized form is obtained by using the combination of prism with A n and T n , that is discussed and is given in Figure 1.E n consists of n-, 5-, 4-, and 3-sided faces.Te desired fgure consists of various cycles induced by 2 Journal of Mathematics vertices as frst cycle is , and the generalized way is in Figure 1.Te theorem given in the following is for the bound of pd(E n ), where four sets of vertices are enough for V(G).

Theorem 3. Let E n be the generalized convex polytope that has
Proof.Te proof of the desired theorem is discussed in the following cases.

□
Case 4. When n � 2ϱ, ϱ ≥ 3, and ϱ ∈ Z + , then the vertices of E n are divided into four partition resolving sets that are For the desired proof, this will be enough to show that the representation of all the vertices of E n is diferent w.r.t partition resolving set Γ and then pd(E n ) ≤ 4. Tat is why, we give the representation of E n w.r.t partition resolving set Γ.

Te representation of vertices of the inner cycle or frst
cycle of E n is given in the following equation: n-1 Te vertices of second cycle of E n have the following representation: Te vertices of third cycle of E n have the following representation: Te vertices of fourth cycle of E n have the following representation: Te vertices of ffth cycle of E n have the following representation: 4 Journal of Mathematics Te vertices of sixth and onward cycles of E n have the following representation, where k ∈ N and k ≥ 6, and show the position of the cycles: Case 5. When n � 2ϱ + 1, ϱ ≥ 3, similarly as in Case 4, the vertices of E n are resolved into four partitions that are Γ � Our aim is to show that the vertices of E n have unique representation w.r.t Γ and then pd(E n ) ≤ 4.
Te following are the representations of vertices of E n w.r.t Γ.
Te vertices of inner cycle of E n have the following representation: Te vertices of second cycle of E n have the following representation:

Journal of Mathematics
Te vertices of third cycle of E n have the following representation: Te vertices of fourth cycle of E n have the following representation: Te vertices of ffth cycle of E n have the following representation: Te vertices of sixth and onward cycles of E n have the representation given in the following equation, where k ∈ N and k ≥ 6: 6 Journal of Mathematics Te graph G n in generalized form has n-, 6-, 5-, 4-, and 3-sided faces [13]; such graph is shown in Figure 2. Te frst cycle consists of vertices u ξ : , similarly the other cycles are in the desired fgure.For the bound of pd(G n ), we represent the following theorem.Te theorem shows that, for the desired purpose, only four sets of vertices are enough for partition of V(G n ).
Theorem .Let G n be the generalized convex polytope with n ≥ 6, then pd(G n ) ≤ 4.
Proof.For the proof, the following cases are discussed.□ Case 7.For n � 2ϱ, with ϱ ≥ 3 and ϱ ∈ Z + .For the desired purpose, the vertices of G n are divided into four sets.Te sets are . Tis will be enough to show that the vertices of G n have unique representation of vertices w.r.t Γ and then pd(G n ) ≤ 4. For this, the following is the representation w.r.t Γ.
Te vertices of frst cycle of G n have the following representation: Te vertices of second cycle of G n have the following representation: Te vertices of third cycle of G n have the following representation: Te vertices of fourth cycle of G n have the following representation: Te vertices of ffth cycle of G n have the following representation: Te vertices of remaining cycles of G n have the representation given in the following equation with k ∈ N and k ≥ 6: Case 8.
where the vertex set of G n is divided into four sets that are For the desired purpose, we have to show that the vertices of G n have unique representation w.r.t Γ and then pd(G n ) ≤ 4.
Te representations w.r.t Γ are given in the following.Te vertices of frst cycle of G n have the following representation: Te vertices of second cycle of G n have the following representation: Te vertices of third cycle of G n have the following representation: Te vertices of fourth cycle of G n have the following representation: Te vertices of ffth cycle of G n have the following representation: Te remaining cycles of G n have the following representation with k ≥ 6 and k ∈ N: 2.3.Generalized Convex Polytope S n .Te formation of convex polytopes is in [13].S n consists of 3-, 4-, 5-, and n− sided faces.Te arrangement of cycles in S n is like the frst cycle is composed of vertices u ξ : 1 ≤ ξ ≤ n   and the second cycle is composed of vertices v ξ : 1 ≤ ξ ≤ n  .Te general way of arrangement of cycles is shown in Figure 3. Te following theorem is for pd(S n ), which shows that only four sets of vertices are required for the desired purpose.
Theorem 9. Let S n be the generalized convex polytope with n ≥ 6, then pd(S n ) ≤ 4.
Proof.We represent the proof in two cases given as follows.

□
Case 10.For n � 2ϱ, with ϱ ≥ 3 and ϱ ∈ Z + , the vertices of S n are divided into four sets, as shown in Figure 3. Te sets are For the desired proof, this will be enough to show that the vertices of S n have unique representation w.r.t Γ and then pd(S n ) ≤ 4. Te representation is given in the following equation.
Te vertices of frst cycle in S n have the following representation: Te vertices of second cycle of S n have the following representation: Te vertices of third cycle of S n have the following representation: Te vertices of fourth cycle of S n have the following representation: Te representation for the ffth and onward cycles is given in the following equation, where k ∈ N and k ≥ 5:

Conclusion
In this work, diferent types of convex polytopes are considered, and these polytopes are generalized by the addition of some cycles that are discussed in the main work.All the new cycles are generated up to some number; then, general representations were given for representing further cycles.Te polytopes that are discussed for the partition dimension in generalized form are E n , G n , and S n .Also, we obtained the bounds for the partition dimension of the desired polytopes, and the bound for the partition dimension of the considered polytopes is found to be 4 or less [35].