Abstract
The concept of locating number for a connected network contributes an important role in computer networking, loran and sonar models, integer programming and formation of chemical structures. In particular it is used in robot navigation to control the orientation and position of robot in a network, where the places of navigating agents can be replaced with the vertices of a network. In this note, we have studied the latest invariant of locating number known as local fractional locating number of an antiprism based convex polytope networks. Furthermore, it is also proved that these convex polytope networks posses boundedness under local fractional locating number.
1. Introduction
Slater [1] introduced the methodology to compute the locating set of a connected network. He defined the minimum cardinality of a locating set as a locating number of a connected network. Melter and Harary independently studied the concept of location number but they used the different term called as metric dimension. They also briefly studied the locating number of serval type of networks such as cycles, complete and complete bipartite networks [2]. Applications of locating number can be found for navigation of reboots [3], chemical structures [4], combinatorial optimization [5] image processing pattern recognition [6].
Chartrand et al. [4] played a vital role in the study of locating number (LN), they characterized all those connected networks of order having locating number 1, , and . Furthermore, they also presented a new technique to compute bounds of locating number of unicyclic networks. Since then researchers have computed locating number of many connected networks such as generalized Peterson network [7], Cartesian products [8], constant locating number of some convex polytopes and generalized convex polytopes [9, 10], Mobius ladders [11], Toeplitz networks [12], k—dimensional networks, and fan networks[13, 14]. Moreover, LN of corona product and partition dimension of different products of networks can be seen in [15, 16] and fault tolrent LN of some families of convex polytopes studied in [17, 18]. For the study of edge LN of wheel and k level wheel networks, we refer [19, 20]. There are various new invariants of LN which have been introduced in recent times such as partition dimension [21], Strong—LN [5], fault-tolerant LN [22], edge LN [23], mixed—LN [24], independent resolving sets [25], and K—LN [26].
Chartared et al. use the concept of LN to solve an integer programming problem (IPP) with specific conditions [4] and Currie and Olllermann used the idea of fractional locating number (FLN) to find solution of specific IPP as well [27]. The FLN formally introduced in networking theory by Arguman and Mathew and they computed exact values of FLN of a path, cycles, wheels, complete and friendship networks. Furthermore, they also developed some new techniques to compute exact values of FLN of connected networks with specific conditions [28]. Later on Arguman et al. characterized all those networks have FLN exactly and they also presented many results on FLN of Cartesian product of two networks [29]. Feng et al. computed FLN of distance regular and vertex transitive networks [30]. For the study of FLN of corona, lexicographic, and hierarchical products of connected networks see [31, 32]. Recently, Alkhalidi et al. established sharp bound of FLN of all the connected networks [33].
The latest version of FLN called by local fractional locating number (LFLN) is defined by Aisyah et al. and they computed LFLN of different connected networks [34]. Javaid et al. developed sharp bounds of LFLN of all the connected networks and they computed upped bounds of local FLN of wheel-related networks. Furthermore, they also improved the lower bound of LFLN different from unity and also developed a technique to compute exact value of LFLN under specific conditions [35, 36]. For the study of LFLN of generalized gear, sunlet and convex polytope networks see [37–39].
In this manuscript, our main objective is to compute LFLN of cretin family of convex polytopes in the form of sharp upper and lower bounds. It has been proved that in every case the convex polytopes remain bounded. The manuscript is organised as Section 2 contains preliminaries and Sections 3 and 4 have main results and conclusion respectively.
2. Preliminaries
A network is an order pair , where is the vertex set and is the edge set. A walk is a finite sequence of edges and vertices between two vertices. A trail is a walk in which all edges are distinct and a path is a trail in which all vertices are distinct. A network is connected if there is a path between each pair vertices. The distance between two vertices and is donated by is defined as the length of the shortest path between and . For further preliminary results of networking theory see [40]. A vertex is said to resolve a pair , if . Suppose that and , then tuple representation of with respect to is defined as If distinct vertices of have unique representation with respect to then is known as locating/resolving set. The minimum cardinality of is called locating number (LN) of that is defined as
For an edge the local resolving neighbourhood set (LRN) is the collection of all vertices of which resolve an edge and it is donated by , where . A real valued function becomes local resolving function of if for each in , where . A local resolving function (LRF) is called minimal LRF if there exist another function such that and for at least one , that is not LRF of . If , then local fractional locating number (LFLN) of is defined as
3. Main Results
This section is devoted to the main results in which, we have examined the LFLN of cretin family convex polytope networks and and it has been proved that these polytope networks remain bounded under LFLN when their order approaches to infinity.
3.1. LFLN of Convex Polytpoe Network
In this particular subsection, we have computed resolving local neighbourhood sets and LFLN of in the form of exact values and sharp lower and upper bounds.
The convex polytope network is introduced by Bača [41] and LN of is 3 is proved in [10]. The vertex set consists of inner , middle , and outer vertices . The edge set is defined as . Furthermore, the order and size of are and respectively and for complete details see Figure 1.
Lemma 1. Let be a convex polytope network, with and . Then.(i) and .(ii) and .
Proof. Consider inner, middle and are outer vertices of , where and .(i) and . Note that and .(ii), , and .The cardinalities of all the LRN sets of are illustrated in Table 1.
From Table 1, we note that . Since therefore .
Theorem 1. Let be a convex polytope network. Then
Proof. The LRN sets of convex polytope network are: ,Since, which is less then the other LRN sets, where . Furthermore, and .
Therefore, we define an upper LRF as . In order to show that is minimal LRF consider another upper LRF as therefore and which shows that is not LRF of hence . Likewise for cardinality of LRN set is 10 which is greater then the cardinalities of all other LRN sets of . Therefore, we define a maximal lower LRF as . Hence . Consequently,
Theorem 2. Let be a convex polytope network. Then
Proof. The LRN sets are given by: ,It is clear that the cardinality of each RLN set of is 16. Therefore, we define a constant function as . Hence .
Theorem 3. Let be a convex polytope network, with and . Then
Proof. To prove the result, we split it into two cases
Case 1. For , we have following LRN sets;,Since, and , where
Furthermore, and . Hence, we define an upper LRF as . In order to show that is minimal upper LRF consider another mapping as therefore and which shows that is not LRF of hence . Likewise for cardinality of LRN set is 24 which is greater then the cardinalities of all other RLN sets. Therefore, we define a lower LRF as is a maximal lower LRF hence . Consequently,
Case 2. For , by Lemma 1, and . Furthermore, . Hence an upper LRF is defined as . In order to show that is minimal upper LRF consider another function as therefore and which shows that is not LRF of hence . Likewise the cardinality of RLN set is which is greater then the cardinalities of all other LRN sets. Therefore, we define a maximal LRF as , hence .
Consequently,
Lemma 2. Suppose that is a convex polytope network, with and . Then.(i) and .(ii) and .
Proof. Consider inner, middle and are the outer vertices of , where and .(i) and . Note that and .(ii), , , . The cardinalities of each LRN set of is illustrated in Table 2.It is can be observed with the help of Table 2 that . Since therefore .
Theorem 4. Let be a convex polytope network, with and . Then
Proof. In order to prove the result, we split into two cases:
Case 3. For , we have following LRN sets; Since, and , where Furthermore, and . Hence, we define an upper LRF as . In order to show that is minimal upper LRF consider another function as therefore and which shows that is not LRF of therefore . Likewise for cardinality of LRN set is 13 which is greater then the cardinalities of all other LRN sets. Therefore there exist a maximal lower LRF and which is defined as hence . Consequently,
Case 4. For , by Lemma 2, and . Furthermore, . Hence there exist an upper LRF and is defined as . In order to show that is minimal upper LRF consider another function as therefore and which shows that is not LRF of hence . Likewise the cardinality of LRN set is which is greater or equal to the cardinalities of all other LRN sets of . Hence, we define a maximal lower LRF is as , therefore .
Consequently,
3.2. LFLN of Convex Polytpoe
In this particular subsection, we have computed the LRN sets and LFLN of convex polytope network . The and . The order and size of is and respectively. For more details see Figure 2.
Lemma 3. Let be a convex polytope network, with and . Then.(i) and ,(ii) and .
Proof. Consider inner, middle and are outer vertices of , where and .(i) and . Note that and .(ii), , , , Now, we illustrate the cardinalities of the LRN sets in Table 3 and also compare them.It can be observed with the help of Table 3 that . Since therefore .
Theorem 5. Let be a convex polytope network. Then
Proof. The LRN for convex polytope areFor the cardinality of each LRN set is 8 which is less then the other LRN sets of . Furthermore, and .
Hence there exist an upper LRF is defined as . In order to show that is minimal upper LRF consider another function as therefore and which shows that is not LRF of hence . Likewise for cardinality of LRN set is 12 which is greater then the cardinalities of all other LRN sets. Hence there exist a maximal lower LRF is defined as , therefore . Consequently,
Theorem 6. Let be a convex polytope network, with and . Then
Proof. In order to prove the result, we split it into two cases
Case 5. For , we have the following LRN sets;Since, and , where . Furthermore, and . Hence, we define an upper LRF as . In order to show that is minimal upper LRF consider another function as therefore and which shows that is not LRF of therefore LFLN . Likewise for cardinality of LRN set is 20 which is greater then the cardinalities of all other LRN sets. Hence there exist a maximal lower LRF and it is defined by , therefore . Consequently,
Case 6. For , by Lemma 3, and . Furthermore, . Therefore, we define an upper LRF as . In order to show that is minimal upper LRF consider another function as therefore and which shows that is not LRF of hence . Likewise the cardinality of LRN set is which is greater then the cardinalities of all other LRN sets of . Hence there exist a maximal LRF and it is defined as therefore . Consequently,
Lemma 4. Let be a convex polytope network, with and . Then.(i) and .(ii) and .
Proof. Consider inner, middle and are outer vertices of respectively, where and .(i) and . Note that and .(ii), , , , , The cardinalities of all the LRN sets are illustrated in Table 4.Now it is clear that . Since therefore .
Theorem 7. Let be a convex polytope network, where and . Then
Proof. In order to prove the result, we split it into two cases
Case 7. For , we have following LRN sets;Since, and , where Furthermore, and . Hence, we define an upper LRF as . In order to show that is minimal upper LRF consider another function as therefore and which shows that is not LRF of therefore . Likewise for cardinality of LRN set is 20 which is greater then the cardinalities of all other LRN sets. Hence there exist a maximal lower LRF is defined by , therefore . Consequently,
Case 8. For , by Lemma 4, and . Furthermore, . Hence, we define an upper LRF as . In order to show that is minimal upper LRF consider another function as this implies and which shows that is not LRF of therefore . Likewise the cardinality of LRN set is which is greater then the cardinalities of all the other LRN sets. Therefore there exist a maximal lower LRF and it is defined as therefore . Consequently,
4. Conclusion
In this dissertation, we studied the LFLN of different families of convex polytope networks (, ) and after establishing the bounds of LFLN of both convex polytope networks, we conclude that both of them posses boundedness when .
Exact value of LFLN in one case is,(i).(ii)Boundedness of LFLN of and illustrated in Table 5.
Now, we close our discussion with the following open problem, characterize all the classes of convex polytopes networks those attain exact value of local fractional locating number.
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 they have no conflicts of interest regarding this article.
Acknowledgments
The authors are deeply indebted to the anonymous referees for their valuable thoughts and comments which improved the original version of this paper. The first author (Hassan Zafar) and the second author (Muhammad Javaid) are supported by the Higher Education Commission of Pakistan through the National Research Program for Universities Grant NO. 20-16188/NRPU/R&D/HEC/2021 2021. There is no source of funding for this submission.