Abstract
Metric dimension is the extraction of the affine dimension (obtained from Euclidean space ) to the arbitrary metric space. A family of connected graphs with is a family of constant metric dimension if (some constant) for all graphs in the family. Family has bounded metric dimension if , for all graphs in . 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.
1. Introduction
Let be a finite, simple, and undirected connected graph with vertex set and edge set . The distance between two vertices is denoted by where is the length of the shortest path between these vertices in . Moreover, the distance because all graphs are undirected. An ordered subset of is called a resolving set or locating set for if for any two distinct vertices and , their codes are distinct with respect to , where is a vector [1]. is called the metric dimension or locating number of , and such a resolving set is called a basis set for . To investigate is a basis set for , it suffices to show that, for all different vertices , their codes are also different because for any , the component of the code is zero, while all other components are nonzero. For more details about and resolving sets, one can read [1–4].
Lemma 1 (see [3]). For a connected graph with resolving set , if for all , then .
The join of two graphs and represented as is a graph with and . is a wheel graph of order for . is a fan graph obtained from the amalgamation of the path on vertices with a single vertex graph . Jahangir or gear graph is obtained from the wheel graph by deleting -cycle edges alternatively; see in [4]. The following results appear in [5–7] for the graphs defined above.
Theorem 1. For wheel graph , fan graph , and Jahangir graph , we have the following:(i), for every (ii), for every (iii), for every
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 , where ; then, the following holds:(i)There is a unique shortest path between and (ii) or (iii)For every other vertex except and on ,
Definition 1 (see [11]). A set is said to be convex if the line segment , lies inside for all distinct pairs of point .
Definition 2 (see [11]). The smallest convex set containing (the intersection of the family of all convex sets that contain ) is called the convex hull of , denoted by Conv, where is a convex set.
Definition 3 (see [11]). A convex polytope is a bounded convex linear combination of convex sets.
There are some families of graphs with constant metric dimension (see [2]); these families are generated by convex polytopes. The problem of finding is NP-complete (see [2]).
Theorem 3 (see [12]). Let be a convex polytope with -pendent vertices; then, for all .
Theorem 4 (see [12]). The metric dimension of convex polytope with -pendent edges is 3 for every .
Theorem 5 (see [12]). for , where is a convex polytope graph with -pendents.
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.
2. Main Results
This section is devoted to the main results which we proved for the newly introduced generalized convex polytopes. The convex polytopes , , and 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 is the generalization of , with one -sided and infinite face each, 3-sided faces being , and 4-sided faces being , so the total number of faces is . The convex polytope is obtained from the generalized convex polytope graph by attached -pendent vertices at the outer cycle of , shown in Figure 1. The generalized convex polytope with -pendents is a graph consisting of cycles, with vertex and edge sets
In the set of edges, indices are taken as modulo and .
In [2], it was shown that , for . In the result below, we proved that the metric dimension for the generalized convex polytope of is still 3, which implies that , , and belong to the same family of constant metric dimension.
Theorem 6. Let be the generalized convex polytope graph defined above; then, for and .
Proof. Validating the mentioned theorem with the help of double inequalities, two cases are present: Case (i): for is even. Let where is an integer. As for all , it is guaranteed by [15] that . Consider to be an ordered subset of ; to show that is a basis set for , codes of the elements of with respect to are given in the following scheme: Codes for the vertices for and are given in the following: It proves that implies that the metric dimension of is 3. Case (ii): for is an odd integer. Let , where , and by [15], ; for reaching the conclusion, it remains to show that . Let be an ordered subset of ; the formulation for the representation of nodes for with respect to is given in the following:The representation of the vertices and , is as follows:It shows that, for any two distinct vertices for odd , implying that ; this completes the proof.
3. Generalized Convex Polytope Graph
In [16], Imran et al. proved the metric dimension of convex polytope . The general form of is denoted by known as the generalized convex polytope (for short, GCP); this graph consists of one each -sided and infinite face, respectively, and the number of 3-sided faces is and 4-sided faces is . The GCP graph is a graph with -pendent edges. Vertex and edge sets for are given in the following:
In the set of edges, indices are taken as modulo and . In Figure 2, the graph is shown.
The result given below shows that belongs to the family of constant metric dimension.
Theorem 7. Let be a GCP graph with -pendents for all ; then, .
Proof. As for all nonpendent vertices of , for all by [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 is an even integer. Let with ; consider an ordered subset of vertices of . The representation of vertices of with respect to is formulated as follows: For , Codes of the pendent vertices are given as follows: From the above formulation, it is obvious that no two distinct vertices of the GCP with pendents have the same code with respect to , which implies that . Case (ii): for is an odd integer. Let for ; suppose an ordered subset of vertices ; to show that is a basis set for , the formulation codes are given as follows:Codes given to the vertices of other interior cycles areRepresentation given to the second last cycle isThe same representation is given to the pendent vertices:It shows that is a resolving set for for -odd and -pendents, , and this completes the proof.
4. Generalized Convex Polytope Graph
In [2], the graph is given, and a generalized graph of is shown in Figure 3. The vertex and edge sets for this graph are given as follows:
We will show that GCP graph with along with -pendent vertices belongs to the family of constant metric dimension and its locating number is 3.
Theorem 8. Let be a GCP graph for ; then, .
Proof. According to [15], if and only if or for all as for every nonpendent vertex of implying that . To reach the conclusion, it remains to show that there exists a resolving set for with exactly three elements. For this, consider the following two cases:
Case (i): for an integer is even. Let , where ; take to be an ordered subset of ; to show that resolves vertices of the GCP, the representation of vertices of the GCP is shown as follows:For ,
Representation given to the cycle is
Representation of the vertices of the outer cycle is
Representation of pendent vertices is
It shows that is a resolving set for GCP implying that .
Case (ii): when is an odd integer. Let for ; let be an ordered subset of the vertices of the GCP. To show that is a locating set for , consider the codes’ formulation of the vertices of the GCP with respect to as
For ,
Representation given to the vertices of the interior cycle is
Representation given to the nodes of the outer cycle is
Also, representation given to the nodes hanging is
It gives us that is a resolving set for GCP , implying that the metric dimension of GCP is 3.
5. Concluding Remarks
In this paper, we focus to study those graphs obtained from convex polytopes and examine that generalized convex polytopes (GCPs) also belong to the family of constant metric dimension such as their parent graphs , , and .
Data Availability
The data used to support the findings of this study are cited at relevant places within the text as references.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Authors’ Contributions
All authors contributed equally to this work.