Abstract
Let be a graph with vertices and is an -clique of . A vertex is said to resolve a pair of cliques in if where is the distance function of . For a pair of cliques , the resolving neighbourhood of and , denoted by , is the collection of all vertices which resolve the pair . A subset of is called an -clique metric generator for if for each pair of distinct -cliques and of . The -clique metric dimension of , denoted by , is defined as is an -clique metric generator of . In this paper, the -clique metric dimension of corona and edge corona of two graphs are computed. In addition, an integer linear programming model is presented for the -clique metric basis for a given graph and its -cliques.
1. Introduction
Throughout this paper all graphs are assumed to be finite, simple, connected, and undirected. For a positive integer number , we use the notation instead of . A clique is a collection of vertices of a graph in which every two distinct vertices are adjacent. An -clique is a clique with vertices. For an -clique of a graph , we will also use the notation to denote . For a vertex and an -clique of a graph , notation denotes where is as usual the number of edges on a shortest -path.
Let be a graph with vertices and is an -clique of . For a pair of cliques , the resolving neighbourhood of and , denoted by , is the collection of all vertices which resolve the pair . A subset of is called an -clique metric generator (-CMG for short) for if for each pair of distinct -cliques and of . A -clique metric generator of minimum cardinality is called an -clique metric basis of . The cardinality of an -clique metric basis of , denoted by , is said to be -clique metric dimension (-CMD for short) of .
Here, -CMD is considered as a generalization of the concept of a -metric dimension presented in [1]. Indeed, -CMD is known as -metric dimension and is denoted by in [1]. In addition, (1,1)-CMD is known as metric dimension which is the first version of this type of invariants (see [2] for more details). After that, other versions of metric dimension such as edge metric dimension and mixed metric dimension were also defined (see [3–6] for more information about these topics). In what follows, we will also use notation instead of .
In the next section, we need an extension of the concept -metric dimensional of graph defined in [7] as follows.
A graph is -clique metric dimensional if is the largest integer such that there exists a -clique metric generator for .
Consider graph shown in Figure 1. We want to compute . Then first, we find for each pair of distinct 2-cliques (edges) and of . , , , , , , , , . According to , , , and , we can conclude that vertices must be the members of each (2,2)-clique metric generator of . Therefore, is a (2,2)-clique metric basis of and so .
As another example, we compute of graph depicted in Figure 1. This graph has two 3-cliques and . Thus, . Then is a (3,2)-clique metric basis of and so .
Lots of work have been done in -metric generator sets of graphs. We recommend [1] for more details on this topic. Estrada-Moreno et al. studied -metric dimension of corona product graphs in [8]. In this paper, we give and in terms of the global forcing numbers of , the order, and size of . We also present an integer linear programming model for the -clique metric basis for a given graph and its -cliques.
2. Main Results
To state our main results, we need to introduce the concept of -global forcing set for -cliques as an extension of the idea of global forcing sets for -cliques of a graph which was presented in [9].
Let and be two positive integer numbers. A -global forcing set for -cliques of a graph is a subset of with this property that for any two distinct -cliques and of . A -global forcing set for -cliques of with minimum cardinality is called a minimum -global forcing set for -cliques of , and its cardinality, denoted by , is called the -global forcing number for -cliques of .
For finding a global forcing set for -cliques of , an ILP model was presented in [9]. We extend this model to achieve the following ILP for finding a -global forcing set for -cliques of .
Let be a graph with and let be the set of all -cliques of . Suppose that is a matrix, where if , and otherwise. The goal is to minimize subject to the constraints:(i).(ii) with if vertex and otherwise.
It is not difficult to see that if is a set of values for which attains its minimum, then for is a minimum -global forcing set for -cliques of . In addition, .
2.1. -CMD of Corona Product
Let and be two graphs with . The corona product is obtained from one copy of and copies of by joining with an edge each vertex of the th copy of , , to , see [10]. In this subsection, , , denotes the th copy of in .
Theorem 1. Let be a graph with vertices and be a graph with more than one -clique. Then
Proof. Let be a -global forcing set for -cliques of that . Set . Obviously, . We claim that is an -CMG of . To prove our claim, we investigate the following cases for -cliques and in . Case 1. and are two distinct -cliques of for an . Then and consequently . Case 2. and are two distinct -cliques of for an . Then there exist -cliques and in such that and . Thus, . This concludes that and so . Case 3. and do not satisfy in Case 1 and Case 2. In this case, it is not difficult to check that there exists , , such that . Hence, and so .According to cases 1–3, is an -CMG for which implies that . Then it is sufficient to prove that .
Let be an -clique metric basis of . Thus, it is enough to prove that is a -global forcing set for -cliques of . Let and be two distinct -cliques of . Since is an -clique metric basis of , then there exist at least vertices such that for every where and . On the other hand, clearly for each . Thus, we deduce that . Thus, is a -global forcing set for -cliques and . Therefore, .
Consider graph shown in Figure 2. In this figure, 3-cliques are as , , , . Hence, . By a similar argument, we have , , , , and . Since , then . Now, we are ready to compute by the previous theorem. Clearly, . Therefore, .
Theorem 2. Let be a graph and with . Then is (2,2)-clique metric dimensional and
Proof. First, we show that is (2,2)-clique metric dimensional. In other words, we prove that has no -clique metric generator if . For this aim, we show that for any . Suppose and . Thus, and so .
Now, according to Theorem 1, we have . On the other hand, it is not difficult to check that . Therefore, .
2.2. -CMD of Edge Corona Product
Let be a graph of size and be a graph. The edge corona product of graphs and is obtained from one copy of and copies of by joining with an edge each vertex of the copy of , , to vertices of the edge of , cf. [11]. If , then the copy of in corresponding to the edge of will be denoted with .
Theorem 3. Let be a positive integer number and be a graph with more than one -clique. If is a graph of size without any pendant vertices such that for every two 2-cliques and in and for every two -cliques and in , then
Proof. Suppose that is a -global forcing set for -cliques of that and set . To achieve , we prove that is a -CMG of . To do this, we investigate below cases for two distinct -cliques and of . Case 1. and are -cliques of for an . Then there exist -cliques and in such that and . Thus, . Hence, and so . Let . A similar argument shows that where and are -cliques of , or and are -cliques of , for an . Case 2. and are -cliques in and where and . Clearly there exist two -cliques and in such that . Then . Hence, and so . Case 3. and do not satisfy in Case 1 and Case 2. In this case, one can check that there exists such that . Then which concludes .Therefore, is a -CMG for .
Now we prove . Let be -clique metric basis of . It is enough to prove is a -global forcing set for -cliques of , for . Suppose that and are two distinct -cliques of , for . Let . Since is a -clique metric basis of , then there exist at least vertices such that for every where and . On the other hand, clearly for each . Thus, we conclude that . Hence, is a -global forcing set for -cliques and in . Therefore, .
Consider shown in Figure 3. , , , , , , , , , , , , , , , are 3-cliques of . Then , . In addition, by similar argument, we have , , , , , , , , , , , , , , , . By a similar method, one can obtain other . Since , then . Now, we are ready to compute by the previous theorem. Clearly, . Therefore, .
Theorem 4. If is a nontrivial graph is a graph of order with this property that for every two 2-cliques and in . Then is a (2,2)-clique metric dimensional and
Proof. In order to show that is a (2,2)-clique metric dimensional, we need to prove for any . Suppose and . Thus, and so .
Now, let be a (2,2)-clique metric basis of . Assume, to the contrary, that there exists for an . Then where is an end point of , which is a contradiction. This concludes that . On the other hand, obviously is a (2,2)-clique metric generator of and so . Therefore, .
2.3. Integer Linear Programming Model
In [9], Afkhami et al. gave an integer linear programming model (ILPM) to deal with the -clique metric dimension. Motivated by this work, we here present an ILPM for the -clique metric basis for a given graph and its -cliques. Let be a graph with . Suppose that is the set of all l-cliques of . In addition, suppose that is a matrix such that and . For , , define . The goal is to minimize subject to the constraints
Clearly, if is a set of values for which is attained, then is a -clique metric basis for .
3. Application of -Clique Metric Generator in Self-Driving Car Navigation
A self-driving car needs to determine its position on the city’s streets uniquely. In other words, each street of the city needs code which uniquely determines its location. Therefore, if we consider the city as a graph that edges of are corresponding to the city’s streets, then an edge metric generator of would be the codes of streets. We note that a self-driving car calculates its location by measuring the distance to a set of landmarks placed in certain vertices. In this case, if there are two positions which are only distinguished by a single landmark and communication with this landmark is lost, then the self-diving car cannot find its position. To fix this problem, we have to improve the accuracy of the detection or the robustness of the system. To do this, we should have a family of detectors, say detectors, such that every pair of edges is distinguished by them.
4. Concluding Remarks
-Clique metric dimension of a graph is a parameter that is difficult to compute and that frequently arises in applications. In the present work, we have studied its behavior under corona and edge corona products. It would be of interest to investigate this invariant under other products of graphs such as Cartesian product, lexicographic product, and strong product. We have also presented an integer linear programming model for finding -clique metric dimension of a graph. Then, another interesting thing would be to apply heuristic methods like greedy algorithms, local search algorithms, or metaheuristic algorithms (e.g., simulated annealing and genetic algorithms) for finding near-optimal solutions efficiently.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This research was partially supported by the Ferdowsi University of Mashhad.