Abstract
Metric dimension is one of the distance-based parameter which is frequently used to study the structural and chemical properties of the different networks in the various fields of computer science and chemistry such as image processing, pattern recognition, navigation, integer programming, optimal transportation models, and drugs discovery. In particular, it is used to find the locations of robots with respect to shortest distance among the destinations, minimum consumption of time, and lesser number of the utilized nodes and to characterize the chemical compounds having unique presentation in molecular networks. The fractional metric dimension being a latest developed weighted version of the metric dimension is used in the distance-related problems of the aforementioned fields to find their nonintegral optimal solutions. In this paper, we have formulated the local resolving neighborhoods with their cardinalities for all the edges of the convex polytopes networks to compute their local fractional metric dimensions in the form of exact values and sharp bounds. Moreover, the boundedness of all the obtained results is also proved.
1. Introduction
There are various important tools in the subject of graph theory that are used to model real life problems such as school bus routing, networks of communication, allocation of the frequencies to the radio stations, and social networks consisting of establishing and working relationships between people. Slater formally defined the concept of resolving (locating) sets for a connected network and designed an algorithm to compute them [1]. Harary and Melter also studied the resolving sets of a connected network, and they called the minimum cardinality of a resolving set as metric dimension (MD) [2]. Gary and Johnson proved that the formulating of MD in its general form for all the connected networks is an NP complete problem that leads to characterize the certain families of networks with fixed MD [3]. Tomescu and Melter applied a metric basis in digital geometry and derived some fundamental results about the existence of the rectangles having minimal metric bases of any size greater or equal to three [4].
Chartrand et al. [5] used MD to solve integer programming problem (IPP) with specific conditions. They defined independence resolvent set and established the metric independence number for different families of connected networks. In particular, they also established bounds of the MD of unicyclic networks and proved that MD of a connected network is 1 iff , where (path). Metric dimension generalized Peterson multinetworks are studied in [6], and bounded metric dimension of generalized Peterson network is computed in [7]. After that, local MD [8], strong MD [9], -metric dimension [10], and sharp bounds of partition dimension of convex polytopes are studied [11] in the form of different mathematical expressions and computing algorithms. Moreover, constant MD of some generalized convex polytopes is studied by Zuo et al. [12], and unbounded MD of splitting networks of a path and cycle is computed by Pan et al. [13]. For further study of MD and its applications, we refer [14–16].
The fractional version of MD known as fractional metric dimension (FMD) is introduced by Currie and Oellermann, and they found the nonintegral solutions of IPP by using it [17]. Fahar et al. proposed the optimal solution of linear relaxation of the IPP by using FMD [18]. Arguman and Mathew formally defined the concept of FMD and introduced many basic properties of FMD about the certain families of the connected networks [19]. The bounds of FMD of all the connected networks are established by Alkhalidi et al. [20]. In particular, the FMDs of generalized Jahangir, trees, and unicyclic networks are computed in [21, 22]. Recently, Javaid et al. [23] characterized all the networks with FMD exactly one. The bounds of FMD of metal organic networks and cartesian product of connected networks are computed in [24, 25].
Aisyah et al. [26] introduced the idea of local fractional metric dimension (LFMD), and they also calculated the exact values of LFMD for corona product of connected networks. For all the connected networks, Javaid et al. [27] established upper bound of LFMD. Furthermore, Javaid et al. [28] improved the lower bound of LFMD from unity and illustrated this result by computing the LFMD for the prism-related networks.
Now, we present applications of metric-based dimensions in different fields. In chemistry, the various structures of chemical compounds are considered as the sets of functional groups, where atoms and bonds are presented by vertices and edges, respectively. By permuting the positions of the sets of functional groups, different common substructures are characterized. The concept of resolving sets determines the particular position when two compounds share the same functional groups. This investigation plays a supreme role in drug discovery and pharmaceutical activities. On the other hand, MD is an essential tool to operate movement of navigating agents (reboots) in large sets of landmarks where the resolving sets uniquely determine the positions of navigating agents in the graph space. The problem of minimum machines (robots) to be placed at certain nodes to trace each and every node exactly once is worth investigating in sonar and loran stations. For more details, we refer to [3, 29].
In this paper, we computed LFMD of some families of convex polyposes networks in the form of exact values and sharp bounds. It is also obtained that in all the cases, LFMD of convex polytopes remains bounded. The remaining manuscript includes Sections 2–4 consisting of preliminaries, main results, and conclusion, respectively.
2. Preliminaries
Let be a network with as vertex-set and as edge-set. A path is a simple network in which vertices can be ordered such as two vertices are adjacent if and only of they are consecutive along the list. A network is connected if each pair of vertices of leads a path. For any two vertices , the distance is the number of edges in a shortest path between them. For further study of preliminary concepts of the subject graph theory, see [30].
For a connected network , a vertex resolves if . Let and , then -tuple representation of with respect to donated by is defined as . If the distinct vertices of have different representations with respect to , then is called a resolving set of . The resolving sets of minimum cardinalities are called metric bases in , and the cardinality of a metric basis is known as metric dimension (MD) of that is defined as follows:
For each pair of two adjacent vertices , the local resolving neighborhood (LRN) set is defined as
A local resolving function (LRF) is a real-valued function such that for each of , where . A LRF of is called minimal if there exists some other function such that and for at least one that is not LRF of . If presents all the minimal resolving functions of , then its LFMD is defined as , where [26].
The networks of convex polytopes are fundamental geometric objects that have been investigated since antiquity. They are playing an important role in mathematical subjects like algebraic geometry, combinatorial optimization, and graph theory. For a connected network , Baca et al. [29] constructed the convex polytopes network by the combination of prism and antiprism networks, where . It consists of inner , middle , and outer cycle vertices with and . For more details, see Figure 1.
A network of convex polytope is obtained from by inserting new edges , where , , and (see Figure 2). A network of convex polytope is formed by connecting the alternating bands of triangles of two copies of prism networks. It consists of inner , middle , and outer cycle vertices with and . The edge-set of is defined as . For more details, see Figure 3. The network of convex polytope is obtained from by deleting edges as shown in Figure 4.
Now, we define some important results which will be frequently used in the main results as follows.
Theorem 1 (see [27]). For connected network and the local resolving neighborhood (LRN) set of the edge e, if , thenwhere , , and .
Theorem 2 (see [28]). Let be a connected network with be the vertex-set and be the LRN set for the edge of . Then,where and .
Corollary 1 (see [28]). Let be a connected network and be LRN of . If and , then, , and .
3. Main Results
In this section, we investigate boundedness of LFMD of different convex polytopes networks.
3.1. LRN Sets and LFMD of Convex Polytope
This subsection covers the results related to the LRN sets and LFMD of the convex polynope network that is presented by for any integral value .
Lemma 1. Let be a network of convex polytope, where and . Then, for ,(a), , and (b), (c)
Proof. Assume that , , and are inner, middle, and outer vertices of , respectively, where and .(a)Since and , therefore . Also, , which implies that . Now, .(b)The LRN sets other than and are , , and . Now, we arrange the cardinalities of these LRN sets in Table 1. From Table 1, it is clear that , with .(c)Since , therefore .
Lemma 2. Let with be a network of convex polytope, where . Then,(a) and with (b), (c)
Proof. Assume that , , and are inner, middle, and outer vertices, respectively, of , where and .(a)Consider , therefore , and . Also, which implies that . Furthermore, .(b)Other LRN sets are , , and . The cardinalities of these LRN sets are obtained as given in Table 2. From Table 2, it is clear that , .(c)Since , therefore .
Theorem 3. Let with be a network of convex polytope, where . Then,
Proof. To prove the result, we have following cases: Case 1. For , the LRN sets are as follows: As, for , , , . Furthermore, and . There exists an upper LRF defined as . Consequently, by Theorem 1, . Similarly, for , and , . Thus, there exists a lower LRF such that defined as . Consequently, by Theorem 2, . Since is a nonbipartite network, therefore must be greater than 1. Hence, we have Case 2. For , by Lemma 1, and , . Similarly, and , . Therefore, is defined by is an upper LRF and is defined by is a lower LRF. Moreover, is a nonbipartite network. Therefore, by Theorems 1 and 2, we have
Theorem 4. Let with be a network of convex polytope, where . Then,
Proof. To prove the result, we have following two cases: Case 1. For , the LRN sets are as follows: For , and , . Furthermore, and , . Therefore, there exists a lower LRF defined as for each . Consequently, by Theorem 1, . Similarly, for , and , . Thus, there exists an upper LRF defined as for each . Consequently, by Theorem 2, . Furthermore, is a nonbipartite network, so must be greater than 1. Hence, we have Case 2. For , by Lemma 2, for and , . Therefore, defined by , is an upper LRF. Thus, by Theorem 1, the . Similarly, as and , , where . Therefore, defined by is a lower LRF. Thus, by Theorem 2, . Moreover, as is a nonbipartite network, it implies that must be greater than 1. Hence, we have
3.2. LRN Sets and LFMD of Convex Polytope
This subsection deals with the results for the LRN sets and LFMD of the convex polytopes network with .
Lemma 3. Let be a network of convex polytope, where and . Then, for (a) and with (b), (c)
Proof. Assume that , , and are inner, middle, and outer vertices of , respectively, where and .(a)Since , , , and , therefore with . Also, ; therefore, .(b)The LRN sets of the other and are and . Now, we arrange the cardinalities of these LRN set in Table 3. From Table 3, it is clear that , .(c)Since , therefore .
Lemma 4. Let be a network of convex polytope, where and . Then, for (a) and with (b)(c)
Proof. Assume that , , and are inner, middle, and outer vertices of , respectively, where and .(a)Consider , , , , and . Furthermore, and with .(b)The LRN sets of other than and are and . Now, we arrange cardinalities of these LRN set in Table 4. From Table 4, it is clear that .(c)Since , therefore .
Theorem 5. Let with be a network of convex polytope, where . Then,
Proof. To prove the result, we have following cases: Case 1. For , the LRN sets are as follows: For , and . Furthermore, and There exists an upper LRF defined as , . Consequently, by Theorem 1, . Similarly, for , , . Thus, there exist a lower LRF such that is defined as . Consequently, by Theorem 2, . Hence, we have Case 2. For , by Lemma 3, and , . Similarly, and , . Therefore, is defined by , is an upper LRF and is defined by is a lower LRF. Therefore, by Theorems 1 and 2, we have
Theorem 6. Let with be a network of convex polytope, where . Then,
Proof. To prove the result, we have following cases: Case 1. For , the LRN sets are as follows: For , and . Furthermore, and There exists an upper LRF defined as , . Consequently, by Theorem 1, . Similarly, for , and . Thus, there exists a lower LRF such that is defined as . Consequently, by Theorem 2, . Hence, we have Case 2. For , by Lemma 4, and , . Similarly, and , . Therefore, is defined by , is an upper LRF and is defined by is a lower LRF. Therefore, by Theorems 1 and 2, we have
3.3. LRN Sets and LFMD of Convex Polytope
This subsection deals with the results for the LRN sets and LFMD of the convex polytopes network with .
Lemma 5. Let be a network of convex polytope, where and . Then, for (a) and with (b)(c)
Proof. Assume that , , , and are inner, middle, and outer vertices of , respectively, where and .(a)Consider , , and . Therefore, and . Also, .(b)The LRN sets of other than , , , and are , , , and . The cardinalities of these LRN sets are obtained as given in Table 5. From Table 5, it is clear that .(c)Since , therefore .
Lemma 6. Let be a network of convex polytope, where and . Then, for ,(a), , and (b), (c)
Proof. Assume that , , , and are inner, middle and outer vertices of , respectively, where and .(a)Consider , , and . Therefore, and . Also, .(b)The LRN sets other than , , , and are , , , and . The cardinalities of these LRN sets are obtained as given in Table 6. From Table 6, it is clear that .(c)Since , therefore .
Theorem 7. Let with be a network of convex polytope, where . Then,
Proof. To prove the result, we have the following cases: Case 1. For , the LRN sets are as follows: For , and . Furthermore, and . There exists an upper LRF defined as , . Consequently, by Theorem 1, . Similarly, for , and . Thus, there exist a lower LRF such that is defined as . Consequently, by Theorem 2, . Since is nonbipartite network, therefore must be greater than 1. Hence, we have Case 2. For , by Lemma 5, and . Similarly, and , . Therefore, is defined by , is an upper LRF and is defined by is a lower LRF. Therefore, by Theorems 1 and 2, we have
Theorem 8. Let with be a network of convex polytope, where . Then,
Proof. To prove the result, we have following cases: Case 1. For , the LRN sets are as follows: For , and . Furthermore, and . There exists an upper LRF defined as , . Consequently, by Theorem 1, . Similarly, for , and . Thus, there exists a lower LRF such that is defined as . Consequently, by Theorem 2, . Since is nonbipartite network, therefore must be greater than 1. Hence, we have Case 2. For , by Lemma 6, and . Similarly, and , . Therefore, is defined by , is an upper LRF and is defined by is a lower LRF. Moreover, is a nonbipartite network so must be greater than 1. Therefore, by Theorems 1 and 2, we have
3.4. LRN Sets and LFMD of Convex Polytope
This subsection deals with the results for the LRN sets and LFMD of the convex polytopes network with .
Lemma 7. Let be a network of convex polytope, where and . Then, for ,(a) with .
Proof. Assume that , , and are inner and middle vertices, respectively, and are outer vertices, where and :(a), , , , , and , where .
Lemma 8. Let with be a network of convex polytope, where . Then, for ,(a), , and (b)
Proof. Assuming that , , , and are inner, middle, and outer vertices, respectively, for and , we have(a), and ; therefore, , and also . Moreover, , , , and ; therefore, .(b)Since , therefore .
Theorem 9. Let with be a network of convex polytope, where . Then,
Proof. For , the LRN sets are as follows:Each , where . Furthermore, . Therefore, by Corollary 1,For , by Lemma 7, and , where . Hence, from Corollary 1, we have
Theorem 10. Let with be a network of convex polytope, where .
Proof. To prove the result, we have following cases: Case 1. For , the LRN sets are as follows: For , and . Furthermore, . Thus, there exists an upper LRF defined as . Consequently, by Theorem 1, . Similarly, for , and . Thus, there exists a LRF defined as which is a lower LRF . Consequently, by Theorem 2, . Hence, we have Case 2. For , by Lemma 8, and . Furthermore, with and , . Therefore, defined by , is an upper LRF and defined by , is a lower LRF. Therefore, by Theorems 1 and 2, we have
4. Conclusion
In this article, we studied the boundedness of convex polytopes networks with the help of distance-based parameter that is called by local fractional metric dimension (LFMD). The lower bounds of convex polytopes networks are also improved from unity. The LFMD of the convex polytope , where is . Furthermore, other families of convex polytopes such as , , , and attain both upper and lower bounds between 1 and 2, respectively, as their order approaches to .
Now, we close our discussion with an open problem and characterize all classes of connected networks those remain bounded via LFMD when .
Limiting values of convex polytopes networks are illustrated in Tables 7 and 8.
Data Availability
All the data are included within this paper. However, the reader may contact the corresponding author for more details of the data.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding this article.