Abstract
The parameter of distance in the theory of networks plays a key role to study the different structural properties of the understudy networks or graphs such as symmetry, assortative, connectivity, and clustering. For the purpose, with the help of the parameter of distance, various types of metric dimensions have been defined to find the locations of machines (or robots) with respect to the minimum consumption of time, the shortest distance among the destinations, and the lesser number of utilized nodes as places of the objects. In this article, the latest derived form of metric dimension called as LF-metric dimension is studied, and various results for the generalized gear networks are obtained in the form of exact values and sharp bounds under certain conditions. The LF-metric dimension of some particular cases of generalized gear networks (called as generalized wheel networks) is also illustrated. Moreover, the bounded and unboundedness of the LF-metric dimension for all obtained results is also presented.
1. Introduction
Slater defined the idea of the resolving sets for a connected network to find the reference or location number for a connected network [1]. Harary and Melter studied the same concept of location number with the different term called metric dimension (MD) for connected networks [2]. Melter and Tomescu investigated metric basis in digital geometry those often used in city block distance [3], and Slater studied the dominating and reference sets in a network [4]. Garey and Johnson proved that finding MD of all the connected networks in general form is an NP complete problem [5]. Moreover, Kelenca et al. defined the idea of edge MD, they showed that computing edge MD is NP-complete problem [6], and the directed distance dimension of oriented networks is computed by Chattrand et al. [7]. For further studies of MD of Caylay, mobius ladders, lexicographic product, and Toeplitz networks, we refer to [8–11].
The concepts of various metric dimensions used in engineering are robot navigation, image processing, and pattern recognition [12]. It is used to solve problems involving percolation in hierarchical lattice [13] and to study the structural properties of chemical compounds. More importantly, Chartrand et al. used the concept of MD to solve an integer programming problem (IPP) [14]. The generalized Jahangir network is defined by Tomescu and Javaid [15], and they also computed MD for . Furthermore, MD of generalized certain gear networks and was computed by Imran et al. [16].
Recently, Currie and Oellermann defined the concept of fractional metric dimension (FMD) to improve the solution of the linear relaxation of the IPP [17]. Fehr et al. used FMD to obtain an optimal solution of IPP [18]. Arumugam and Matthew calculated the exact values of FMD of some connected networks [19]. The bounds and exact values of FMD of vertex-transitive and distance-regular networks are computed by Feng et al. [20]. Saputro et al. computed the FMD of comb product of connected networks [21]. For further studies of FMD for hierarchical product, trees, and unicyclic and permutation networks, see [22–24]. Raza et al. computed the bounds of FMD of metal organic frameworks [25]. The FMD of generalized Jahangir network for is computed by Liu et al. [26].
The idea of LF- metric dimension is defined by Aisyah et al., and they also computed exact values for LF- metric dimension for the corona product of connected networks [27]. Liu et al. [28] studied the LF-metric dimension of triangular circular, quadrangular circular, and pentagonal circular ladders. Javaid et al. established the criteria to compute the upper bounds of LF-metric dimension of connected networks and demonstrated the main results by using LF-metric dimension such as wheel-related networks, flowers, and antiweb gear networks [29]. Recently, Javaid et al. improved the lower bound of LF-metric dimension from unity and established the sharp bounds and computed the exact values of LFMD of some prism-related networks [30].
In this article, we have computed LF-metric dimension of generalized gear networks in the form of exact values and sharp bounds. Furthermore, Sections 2 and 3 consist of preliminaries and LRN sets of , respectively. In Section 4, we have computed LF-metric dimension of generalized gear networks, and Section 5 represents the conclusion and limiting values of LF-metric dimension.
2. Preliminaries
Let be a network with and as vertex and edge set, respectively. A walk is a sequence of vertices and edges such that the edge has end points and for . A path between two vertices and is called a walk if the repetition of vertices does not exist [31]. For any two vertices , the distance is the length of the shortest path between them in . A network is said to be connected if there exists a path between any pair of vertices. A vertex is said to resolve a pair if . Let be an ordered set and , then the m tuple representation of with respect to S is . If the distinct vertices of have distinct representations with respect to , then is called resolving (locating) set. The resolving set with minimum cardinality is called the metric basis of , and the cardinality of metric basis is called metric dimension of defined as
For an edge , the local resolving neighbourhood set of is . A real-valued function such that for each LR(ab) is called a local resolving function (LRF) of , where . A LRF is called minimal if there exists a function such that and for at least one that is not LRF of . LF-metric dimension of is donated by and defined as
Now, we present some important results frequently used in this paper.
Theorem 1 (see [29]). Let be a connected network. If is bipartite network, then .
Theorem 2 (see [29]). Let be a connected network and be a LRN set for some . If , thenwhere , , and .
Theorem 3 (see [30]). Let be a connected network and be the LRN set. Then,
Corollary 1 (see [30]). Let be a connected network, be LRN of , , , and . If and , then
Proposition 1 (see [28]). Let be a connected network . If for all , then .
Now, we define generalized gear network as follows.
Definition 1. Generalized gear network sometimes known as bipartite wheel network is obtained from wheel network. Let be the generalized gear network of order , where and . There are three type of vertices in such as major vertices , minor vertices , where and a central vertex . Moreover, . Central vertex is adjacent to major vertices and contains an outer cycle as shown in Figure 1. For more details of , see [16].
3. LRN Sets of the Generalized Gear Network
The resolving neighbourhood sets for each pair of adjacent vertices are classified.
Lemma 1. Let with be a generalized gear network, where . Then, for , we have(a) and (b), and
Proof. Assume that (major) vertices for and (centre) vertex of such that and .(a)Consider , this implies that , where . Furthermore, .(b)Consider , therefore . Moreover, and .
Lemma 2. Let with be a generalized gear network, where , and . Then, for , we have(a) and (b) and ,
Proof. Assume that (major), (center), and be a minor vertex, respectively, where and . Now, we have,(a)Consider , and (b)LRNs other than areThe cardinality of each LRN set is shown in Table 1.
From Table 1, it is clear that . Furthermore, ; therefore, .
Lemma 3. Let be a generalized gear network, where and . Then, for , we have(a) and (b) and
Proof. Assume that (major), (center) and be the minor vertex, for ,. Now, we have(a), and .(b)LRNs other than are , for , , , , , , and .The cardinality of each LRN set is shown in Table 2.
From Table 2, it is clear that . Furthermore, . Therefore, .
4. LFMD of Generalized Gear Networks
In this section, we compute LF-metric dimension of generalized gear network in the form of exact values and bounds under certain conditions.
4.1. For and
In this particular section, we determine the LF-metric dimension of the generalized wheel network, which is the special case of the generalized gear network. For , let be the set of outer vertices and is the central vertex of .
Theorem 4. Let be a generalized gear network, then .
Proof. For and . The LRN sets are given as follows.
The LRN sets of major vertices to central vertex are as follows: The LRN sets of adjacent pair of major vertices are as follows: From above LRN sets, , where and . Hence, a function is an LRF defined by for each . Therefore, by Proposition 1,
Theorem 5. Let be a generalized gear network, then
Proof. For and , the LRN sets are given as follows.
The LRN sets of adjacent pair of major vertices are as follows: The LRN sets of major vertices with central vertex are as follows: From above LRN sets, , where and . Moreover, ; this implies and . Therefore, a function defined by for each is an upper LRF. Consequently, by Theorem 2, .
From above LRN sets, , where and , . Therefore, a function is defined by which is a lower LRF for each . Hence, by Theorem 3, .
Consequently,
Theorem 6. Let be the generalized gear network, then
Proof. For and , the LRN sets are given as follows:
The LRN sets of major vertices with central vertex are as follows: The LRN sets of major vertices with central vertex are as follows: From above LRN sets, , where . Hence, by Corollary 1,
Theorem 7. Let with be a generalized gear network, then
Proof. To prove the result, we have the following cases: Case 1: for k = 0 and , the LRN sets are given as follows: The LRN sets of major vertices with central vertex are as follows: The LRN sets of adjacent pair of major vertices: From above LRN sets, , where and , . Moreover, and . Therefore, a function defined as an upper LRF function for each and 0 otherwise. Therefore, by Theorem 2, . Since , where and , . Therefore, a function defined by for each is a lower LRF, Consequently, by Theorem 3, . Case 2: for k = 0 and . For : by Lemma 1, and , . Furthermore, . Hence, a function is an upper LRF with minimum cardinality defined as Consequently, by Theorem 2. By Lemma 1, , where and , . Furthermore, ; this implies . Hence, function is lower LRF defined by for each ; therefore, by Theorem 3, . Consequently,
4.2. For and
Theorem 8. Let be a generalized gear network, then .
Proof. Let be the set of major, is the set of minor vertices, respectively, and is the central vertex. For and , the LRN sets are given as follows.
The LRN sets of major vertices to central vertex are as follows: The LRN sets of major to minor vertices are as follows: The LRN sets of adjacent pair of minor vertices are as follows: Since each , where , by Corollary 1, .
Theorem 9. Let , where , be a generalized gear network. Then, for ,
Proof. To prove the result, we have the following cases: Case 1: for, and , the LRN sets are given as follows. The LRN sets of major vertices with central vertex are as follows: The LRN sets of adjacent pair of minor vertices are as follows: The LRN sets of adjacent pair of minor vertices are as follows: From above LRN sets, , where and . Furthermore, and . Hence, a function defined by , if and 0 otherwise is an upper LRF. Consequently, by Theorem 2, . From above LRN sets, , where and . Therefore, there exists a lower LRF A function defined by is an LRF of maximum cardinality for each . Hence, by Theorem 3, . Consequently, Case 2: for , , and , where . By Lemma 2, we have the following:(i)(ii) Therefore, a function is an upper LRF with minimum cardinality defined by Consequently, by Theorem 2, . Since and , . Hence, the function is a lower LRF with maximum cardinality, defined by . Therefore, by Theorem 3, . Consequently,
Theorem 10. Let be a generalized gear network with , , where . Then,
Proof. To prove the result, we have the following cases: Case 1: for, m = 4 and k = 2, the LRN sets are given as follows. Adjacent pair of minor vertices is as follows: Major with central vertex. Major with minor vertex. From above LRN sets, , where and . Furthermore, and . Hence, a function defined by if and 0 otherwise is a lower LRF with minimum cardinality. Consequently, by Theorem 2, . From above LRN sets, , where and . Therefore, a function defined by is a lower LRF with maximum cardinality for each . Case 2: for k = 4 and m = 3, the possible LRN sets are as follows. Central to major vertex. Major with minor vertex. Adjacent pair of internal vertices. LRN sets with minimum cardinality are as follows: From above LRN sets, , where and . Moreover, −1, and this implies and . Hence, a function is defined by which is an upper LRF with minimum cardinality for all and 0 otherwise. Therefore, . From above LRN sets , where and , . Therefore, a function is defined by ; hence, by Theorem 3, . Consequently, Case 3: for and , the possible LRN sets are , where . By Lemma 2 and 3, we have(i) and (ii) Therefore, is an upper LRF defined by Consequently, by Theorem 2, . Case 4: since and , . Therefore, defined by is lower LRF. Hence, by Theorem 3, . Consequently,
4.3. For and
Theorem 11. For , k and is a generalized gear network, and then .
Proof. As there is no cycle of odd length in , where . Therefore, is bipartite network; hence, by Theorem 1, .
5. Conclusion
In this article, we studied the LF-metric dimensions of generalized gear networks and established the sharp lower and upper bounds of LF-metric dimensions and computed the exact values in some cases as well.
Exact values of LF-metric dimensions of some cases of the generalized gear networks are as follows: , and Unboundedness is illustrated in Table 3 Now, this section is closed by raising the following open problem: Investigate the LF-metric dimension of the nonregular networks such as convex polytopes, Toeplitz, and Prism-related networks.
Data Availability
The data used to support the findings of this study are included within this article. However, the reader may contact the corresponding author for more details on the data.
Conflicts of Interest
The authors declare no conflicts of interest regarding this article.