On the Metric Dimension of Generalized Tensor Product of Interval with Paths and Cycles
The concept of minimum resolving set for a connected graph has played a vital role in Robotic navigation, networking, and in computer sciences. In this article, we investigate the values of m and n for which and are connected and find metric dimension in this case. We also conclude that, for each m, we obtain a new regular family of constant metric dimension. We also give a basis for these graphs and presentation of resolving vector in general closed form with respect to the basis.
The concept of metric dimension and resolving sets were initially drafted for the metric spaces in 1953 in , p. 95 but did not receive much attention because of the linear continuum nature of standard Euclidean spaces . Later, Slater in [2, 3] and Harary and Melter in  used these ideas of uniquely determining the location of an intruder in a network and graphs. This idea also paved the way for searching uniquely the receiver of a message on a network. Since then the resolving sets have been widely investigated, see, for instance, [5, 6]. The resolving sets also have useful applications in diverse areas including network discovery and verification , strategies for the mastermind games , and applications to problems of pattern recognition, image processing, and digital geometry . Chartrand et al. discussed resolvability and its connection with metric dimension of graphs in [9, 10]. Similarly, on another node, a real-world problem is the study of networks which is not centrally controlled but rather distributed. So, it becomes relatively difficult and costly to obtain a map of all nodes and the links between them. One possible solution is to view network from various local positions and combine them to obtain a good approximation for the real network, . Metric dimension has some applications in this respect as well. Buczkowski et al. discussed a general idea of -dimensional graphs . First, we introduce general terminology related to a graph.
A graph G is an ordered pair , where is the vertex set and is the edge. The distance between vertices , denoted by , is defined as the length of the shortest path between and , and the diameter of , denoted by , is defined as the maximum distance among all pairs of vertices in . A vertex resolves a pair of vertices if . A set of vertices resolves if each pair of distinct vertices of is resolved by some vertex in . The set is called the resolving set of if it resolves . A resolving set of with the least cardinality is called metric basis of , and this cardinality is the called metric dimension of denoted by . It is a common fact now that is unique although we may have many metric bases. Actually the choice of a particular matric basis leads us towards the unique coordinate representation of all vertices of a graph. For an ordered subset of vertices and a vertex in a connected graph , the representation of with respect to is the ordered -tuple.
A lot of work has already been carried out on computation of metric dimension and its other variants related ideas of graphs. Buczkowski et al.  calculated the dimension of wheel and arrived at the result that for . Caceres et al.  computed the dimension of fan graphs, , and proved that, for . Tomescu and Javaid proved that the dimension of Jahangir graphs to be for . Imran et al. computed the dimension of circulant graphs in . Hussain et al. computed bounds for the metric and partition dimension of generalized Mobius ladders in .
In the abovementioned cases, metric dimension of all graphs depends upon the number of vertices in the graph. In contrast, one is interested in the list of all families of graphs in which metric dimension remains constant which seems to be an important feature of the family. Authors in [17, 18] discussed some families of graphs with constant metric dimension. Another aspect is the metric dimension of a graph that is some kind of product of two other graphs. One is often interested in relating the dimension of the product graphs with dimensions of the component graphs. Cartesian product of two graphs and , denoted by is a graph such that the vertex set of is the Cartesian product as a set, and any two vertices and are adjacent in if and only if either and is adjacent with in or and is adjacent with in . The metric dimension of the Cartesian products of graphs has been studied in . Authors proved that the metric dimension of is tied in a strong sense to the minimum order of a so-called doubly resolving set in . Authors also computed bounds on for many examples of and . Authors in  recursively defined generalized corona product and derived some important results about this product in terms of the dimension of components involved, whereas, in , authors computed strong metric dimension of this corona product. Another product is the tensor product, of graphs and which is a graph such that the vertex set of is the Cartesian product ; any two vertices and are adjacent in if and only if is adjacent with and is adjacent with . For example, the tensor product of with is .
Moradi discussed the tensor product of some graphs in . Jannesari and Omoomi computed metric dimension of lexicographic product of some graphs by introducing a new parameter known as adjacency dimension, . Saputro et al. further extended these results for other graphs in . Javaid et al. computed metric dimension of co-normal product of graphs in .
In this paper, we discuss -copies of tensor product of with and . This type of product is relatively less studied in the literature as compared to the Cartesian product. In the first stage, we decide about the connectedness of this product. At the second step, we compute the metric dimension of the connected graphs among them. We like to remark that metric dimension of the tensor product of two graphs has not been extensively studied over the years.
2. Main Results
We focus first on . We compute minimal resolving set for it allowing as distance between two vertices when there does not exist any path connecting them so they are in different components of a graph. The interesting question is the value of for which the graph is connected. The answer, however, is no in this case. Existence of as any component of resolving vector clearly indicates that the graph is disconnected. Although we cannot talk about the metric dimensions in these cases but cardinality of minimum removability does make sense as it shows the connectedness of a graph. We actually give the closed representation of all vertices with respect to our selected metric basis in each case.
Theorem 1. (i) and is disconnected(ii)Cardinality of the minimal resolving set is
Proof. Let , where are vertices and is the vertex of of Kth component. We show that is a resolving set. For this, we give the representation of any vertex of with respect to :Now, we consider the minimality of the resolving set. If we delete from , then . So, cardinality of minimum recoverability is .
Example 1. If and , then and and resolving vectors with respect to areNow, we turn our attention to the . In contrast to the above result, we have an interesting result here. For and , our graph is connected. Moreover, we also give metric dimension in this case as well.
Theorem 2. (1).(2)Cardinality of the minimal resolving set:(3) is connected iff and .
Proof. (1) Let be set of vertices, where are vertices of and and are adjacent vertices of . We show that is a resolving set. For this, we give the representation of any vertex of with respect to :We see that all the representations are distinct, implying that . Now, we prove that is a minimal resolving set. If we take , we receive two similar resolving vectors: , which indicates dimension is not 1. So, is 2.
(2) Case 1 (when n is odd): we compute the general form of resolving vector in this case.
Let , where are vertices of and and are vertices of . We show that is a resolving set. For this, we give the representation of any vertex of with respect to :where , , and .
We see that all the representations are distinct, implying that . Now, we prove that is a minimal resolving set. If we delete from , we receive two similar resolving vectors: , where appears at position, which shows is the minimal resolving vector.
Example 2. If and , then and . and resolving vectors with respect to are
Remark 1. The above example and picture show that graph is disconnected. This holds generally for this case.
Case 2 (when n is even): let . We show that is a resolving set. For this, we give the representation of any vertex of with respect to . When , , and , When , When , , and , When If we delete from , we receive two similar resolving vectors , where appears at position, which shows is the minimal resolving set.
(3) From the above resolving pattern, we can argue this result by presence of at any place.
Remark 2. The same result can be proved by  by the agreement that the tensor product of two graphs is disconnected when both factors are bipartite and connected.
Example 3. If and , then and and resolving vectors with respect to are
In the end, we like to summarize our main outcomes. First, noticeable point is the disconnectedness of all . The sum and substance of this article is the connectedness of for and and metric dimension in this case is 2. So, we obtain a regular family of graph with constant metric dimension 2, which is decomposable into paths and cycles. From network point of view, only this family is significant. However, for the remaining families, we can find minimum resolving set and its cardinality. We conclude this article with
No data were used to support the findings of the study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.
This study was supported by Key Project of Anhui University Natural Fund: Study on Image Semantic Extraction and Optimization Model Based on Voronoi Graph and Random Graph (KJ2013A327) and Wisdom Classroom Project in the University-Level Quality Engineering of Hefei Normal University: Application of Modern Educational Information Technology (2018zhkt10).
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