Abstract
Graph theory is one of those subjects that is a vital part of the digital world. It is used to monitor the movement of robots on a network, to debug computer networks, to develop algorithms, and to analyze the structural properties of chemical structures, among other things. It is also useful in airplane scheduling and the study of diffusion mechanisms. The parameters computed in this article are very useful in pattern recognition and image processing. A number is referred as distance between an edge and a vertex. implies that two edges are resolved by node . A set of nodes is referred to as an edge metric generator if every two links/edges of are resolved by some nodes of and least cardinality of such sets is termed as edge metric dimension, for a graph . A set of some nodes of is a mixed metric generator if any two members of are resolved by some members of . Such a set with least cardinality is termed as mixed metric dimension, . In this paper, the metric dimension, edge metric dimension, and mixed metric dimension of dragon graph , line graph of dragon graph , paraline graph of dragon graph , and line graph of line graph of dragon graph have been computed. It is shown that these parameters are constant, and a comparative analysis is also given for the said families of graphs.
1. Introduction
For several years, the characteristics associated to graph distances have piqued the interest of various scholars, and one of them, the metric dimension, has recently been the focus of them. The theory of metric dimension was given by Slater in 1975 [1] and this theory was further elaborated as resolving set of graphs by Harary and Melter in 1976 [2]. In 2003, Brigham et al. [3] determined resolving dominating set and resolving domination number in graphs. In 2003, Chartrand et al. [4] studied the independent resolving set in nontrivial connected graphs of order . The order of minimal independent resolving set is known as independent resolving number and is denoted as . In 2003, Saenpholphat and Zhang [5] calculated the connected resolving number of complete graph , star graph , and Cartesian products of . In 2007, Oellermann and Peters-Fransen [6] found the strong metric dimension of graphs and digraphs. In 2010, Okamoto et al. [7] discussed the local metric dimension of graphs and some bounds of it. In 2017, Yero et al. [8] determined the k-metric dimensional graphs. In 2016, Imran and Siddiqui [9] computed the metric dimension of some convex polytopes generated by wheel related graphs. A graph in which all vertices have the same degree is called a regular graph. Ali presented a survey of antiregular graphs [10], and he gathered the known results concerning the antiregular graphs.
There are many applications of this parameter to robot navigation in networks which have been discussed in [11] and applications to chemistry have been discussed in [4, 12]. Furthermore, this issue has certain applications to pattern recognition and image processing difficulties, some of which require the usage of hierarchical data structures [13]. Some interesting connections between metric generators in graphs and the master mind game or coin weighing have been presented in [14]. The metric dimension of infinite graphs was studied in [15], and extremal graphs for metric dimension and diameter were considered in [16].
Let be a simple connected and undirected graph with the vertex set and edge set . The distance between any two vertices is denoted as and is defined as the length of the shortest path between and . The vertex resolves the vertices if . An ordered set of vertices ; the identification of depending on is the order tuple and is written as
If the different vertices of have different codes based on , then is known as a resolving set for . Let of graph be such that , then is a metric basis for the graph and is metric dimension of the graph .
Recently, the idea of edge metric dimension of graph was given by Kelenc et al. [17], and they also presented some results and comparison of metric dimension and edge metric dimension for some well-known families of graphs like path graph , where is number of vertices of graph, cycle graph : , complete graph : , and complete bipartite graph : , where are vertices of each partite set; if is a grid graph with , then . Furthermore, they have raised many open problems linked to the nature of metric dimension and edge metric dimension. In 2018, Liu et al. [18] worked on edge version of metric dimension and calculated edge metric dimension of necklace graph and line graph of necklace graph. In 2021, Deng et al. [19] discussed the edge resolvability parameter for different families of Mobius ladder networks, and they find the exact edge metric dimension of triangular, square, and hexagonal Mobius ladder networks.
The distance between an edge and a vertex is denoted as and is defined as . The vertex resolves the edges if . An ordered set of vertices ; the identification of an edge depending on is the order tuple and is written as
If the different edges of have different codes based on , then is known as an edge resolving set for . Let be a subset of vertex set of graph such that , then is an edge metric basis for the graph and is edge metric dimension of the graph .
Kelenc et al. [20] has presented the idea of mixed metric dimension and denoted it as , and they presented the mixed metric dimension of path graph , , cycle graph , , and complete bipartite graph , where ; otherwise, and grid graph , where . An ordered subset of a vertex set of a graph is called mixed resolving set if it is both vertex resolving set and edge resolving set. Let be a graph and be an ordered subset of vertices of graph . If all vertices and edges of have different codes of representation with respect to the set , then is known as a mixed resolving set for graph . Let be a subset of vertex set of graph such that . Then, is known as a mixed metric basis for , and is a mixed metric dimension of the graph . In 2016, Yero [21] presented some bounds or closed formulae for the (edge, mixed) metric dimension of several families of graphs.
Figure 1 represents graph with , and are vertex set and edge set, respectively. The set is an edge resolving set of . The representation of all edges with respect to is as given in Table 1.
All edges have different representations with respect to , so .
The set is a mixed resolving set of . The representation of all edges and vertices with respect to is as given in Table 2.
All edges and vertices have different representations with respect to , so .
Theorem 1 (see [12]). A connected graph of order has dimension 1 if and only if .
Theorem 2 (see [22]). Let be a dragon graph for and . Then, .
Theorem 3 (see [17]). The edge metric dimension of a graph is 1 if and only if is a path.
Theorem 4 (see [20]). The mixed metric dimension of a graph is 2 if and only if is a path.
2. Results on Dragon Graph
Dragon graph is obtained by identifying vertex of cycle graph with vertex of path graph and is denoted as . Order of dragon graph is . Its vertex set is and edge set is where as shown in Figure 2.
Theorem 5. Let be a dragon graph for and . Then, .
Proof. Since dragon graph is not a path graph, . In this case, is an edge resolving set of . All edges are labeled asThe representations of all edges with respect to are as follows:All edges have distinguished representation, and this fact can easily be verified, which implies that . So, we obtained .
Theorem 6. Let be a dragon graph for and . Then, .
Proof. Since dragon graph is not a path graph, . All edges are labeled asIn this case, is mixed resolving set of . The representations of all vertices with respect to are as follows:For even,For odd,The representations of all edges with respect to are as follows:For even,For odd,From the above representation, it is clear that no two vertices, edges, and an edge or a vertex of have the same representation which implies that . So, we obtained .
3. Results on Line Graph of Dragon Graph
A line graph of can be generated by assuming edge set of as vertex set of , and any two vertices of are adjacent if and only if they are neighboring edges in . Line graph of dragon graph has and as its set of vertices and set of edges, where as shown in Figure 3.
Theorem 7. Let be a line graph of dragon graph for and . Then, .
Proof. Since , it is not a path graph . is resolving set of , and the representations of all vertices with respect to are as follows:All vertices have different representation, which implies that . So, we obtained .
Theorem 8. Let be a line graph of dragon graph for and . Then, .
Proof. Since , it is not a path graph . All edges of are labeled as and .
and .
is an edge resolving set of , and representations of all edges with respect to are as follows:All edges have different representation, which implies that . So, we obtained .
Theorem 9. Let be a line graph of dragon graph for and . Then, .
Proof. Since , it is not a path graph . All edges of are labeled as and .
is mixed resolving set of , and the representations of all edges with respect to are as follows:For even,For odd,The representations of all vertices with respect to are as follows:From the above representation, it is clear that no two vertices, edges, and an edge or a vertex of have the same representation which implies that . So, we obtained .
4. Results on Paraline Graph of Dragon Graph
Paraline graph is the line graph of the subdivision graph of any graph. In subdivision graph , the vertex set is and the vertex of corresponding to the edge of is inserted in the edge of . In line graph , the vertex set is and two vertices of are adjacent if the corresponding edges of are incident. Paraline graph of dragon graph has and as its set of vertices and set of edges, where as shown in Figure 4.
(a)
(b)
(c)
Theorem 10. Let be a paraline graph of dragon graph for and . Then, .
Proof. Since , it is not a path graph . is resolving set of , and the representations of all vertices with respect to areAll vertices have different representation, and this fact can easily be verified, which implies that . So, we obtained .
Theorem 11. Let be a paraline graph of dragon graph for and . Then, .
Proof. Since , it is not a path graph . All edges of are labeled as and .
and .
is an edge resolving set of , and representations of all edges with respect to are as follows:All edges have different representation, which implies that . So, we obtained .
Theorem 12. Let be a paraline graph of dragon graph for and . Then, .
Proof. Since , it is not a path graph . All edges of are labeled as and .
is mixed resolving set of , and the representations of all edges with respect to are as follows:The representations of all vertices with respect to are as follows:From the above representation, it is clear that no two vertices, edges, and an edge or a vertex of have the same representation which implies that . So, we obtained .
5. Results on Line Graph of Dragon Graph
Let be a graph and be line graph of line graph of dragon graph . Vertex set of line graph of line graph of dragon graph is , and its edge set is where as shown in Figure 5.
Theorem 13. Let be a line graph of line graph of dragon graph for and . Then, .
Proof. Since , it is not a path graph . is resolving set of , and the representations of all vertices with respect to are as follows:For even,For odd,All vertices have different representations, which imply that . So, we obtained .
Theorem 14. Let be line graph of line graph of dragon graph for and . Then, .
Proof. Since , it is not a path graph . All edges are labeled as and . is an edge resolving set of , and the representations of all edges with respect to are as follows:For even,For odd,All edges have different representations, which implies that . On the other hand, we have to show that .
Suppose to the contrary that , then we have the following possibilities. If the set where and is an edge resolving set for graph , then some edges have same representations as shown in Table 3 and in Table 4.
If the set where and is an edge resolving set for graph , then some edges have same representations as shown in Table 5.
If the set is an edge resolving set for graph , then some edges have same representations as shown in Tables 6 and 7.
The set is not an edge resolving set for graph because it did not resolve the following edges:Any combination of three vertices will not resolve the all edges of the graph . Hence, there is no edge resolving set with three vertices for which implies that . So, we obtained .
Theorem 15. Let be line graph of line graph of dragon graph for and . Then, .
Proof. Since , it is not a path graph . All edges are labeled as and .If is mixed resolving set of , then the representation of all edges with respect to is as follows:For even,For odd,The representation of all vertices with respect to is as follows:All vertices and edges have distinguished representations with respect to set , which implies that . On the other hand, we have to show that .
As we know that the edge metric dimension of line graph of line graph of dragon graph is 4, by Theorem [6]. Let and set be mixed resolving set. The set did not resolve all edges and vertices as for and for . That is why we have to include one more vertex in mixed resolving set which implies that .
If the set is mixed resolving set, then it resolves all edges and vertices of . However, for . The set did not resolve all edges and vertices of ; this implies that .
If the set is mixed resolving set of , then ; the set did not resolve all edges of ; this implies that .
If the set is mixed resolving set of , then ; the set did not resolve all edges of ; this implies that .
If the set is mixed resolving set of , then , , and ,; the set did not resolve all edges and vertices of ; this implies that .
It is clear that is also computed that no set of cardinality smaller than five is a mixed resolving set of . So, we obtained .
6. Comparison of Vertex, Edge, and Mixed Metric Dimension of Dragon Graph
On the basis of some facts given in [17], one can easily conclude that these parameters are not comparable because there are some families graph for which , , or . Moreover, from [23], infinite families of graphs for which is satisfied were described, which was an open problem presented in [17], where only one family satisfying this inequality was given. Any mixed metric generator is, by definition, also a metric generator and an edge metric generator. In this sense, the connection [20] follows instantly. It is obvious that the vertex set of any graph forms a mixed metric generator for any graph . In addition, any vertex of , as well as each incident edge with it, has the same distance to the vertex itself. In this respect, a single vertex in cannot constitute a mixed metric generator.
7. Conclusion
In this article, we calculate the exact value of vertex, edge, and mixed dimension for dragon graph, , line graph of dragon graph, , paraline graph of dragon graph, , and line graph of line graph of dragon graph, . The vertex metric dimension for , , , and is constant and same; . The edge metric dimension for , , , and is constant but . The mixed metric dimension for , , , and is constant but . It is already known from [17] that there are graphs for which , , or . Moreover, dragon graph, , line graph of dragon graph, , and paraline graph of dragon graph, , are families of graphs for which but line graph of line graph of dragon graph is .
Data Availability
All data required to prove the results are already included in the manuscript. So, no additional data are associated with this article.
Conflicts of Interest
The authors declare that there are no conflicts of interest.
Authors’ Contributions
All authors contributed equally for the preparation of this article.
Acknowledgments
This research was supported by the UAEU-AUA grants of United Arab Emirates University (UAEU) via Grant no. G00003461.