A Comparative Study of Three Resolving Parameters of Graphs

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. ,e parameters computed in this article are very useful in pattern recognition and image processing. A number d(f, w) � min d(w, t), d(w, s) { } is referred as distance betweenf � ts an edge andw a vertex. d(w, f1)≠d(w, f2) implies that two edges f1, f2 ∈ E are resolved by node w ∈ V. A set of nodes A is referred to as an edge metric generator if every two links/edges of Γ are resolved by some nodes of A and least cardinality of such sets is termed as edge metric dimension, edim(Γ) for a graph Γ. A set B of some nodes of Γ is a mixed metric generator if any two members of V∪E are resolved by some members of B. Such a set B with least cardinality is termed as mixed metric dimension, mdim(Γ). In this paper, the metric dimension, edge metric dimension, and mixed metric dimension of dragon graph Tn,m, line graph of dragon graph L(Tn,m), paraline graph of dragon graph L(S(Tn,m)), and line graph of line graph of dragon graph L(L(Tn,m)) have been computed. It is shown that these parameters are constant, and a comparative analysis is also given for the said families of graphs.


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. e 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 n. e order of minimal independent resolving set is known as independent resolving number and is denoted as ir(Γ). In 2003, Saenpholphat and Zhang [5] calculated the connected resolving number of complete graph K n , star graph K 1,n−1 , and Cartesian products of Γ × K 2 . 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.
ere 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]. e metric dimension of infinite graphs was studied in [15], and extremal graphs for metric dimension and diameter were considered in [16].
Let Γ be a simple connected and undirected graph with the vertex set V(Γ) and edge set E(Γ). e distance between any two vertices u, v ∈ V(Γ) is denoted as d (u, v) and is defined as the length of the shortest path between u and v. e vertex t resolves the vertices u and v if d(t, u) ≠ d(t, v). An ordered set of i vertices R � x 1 , x 2 , . . . , x i ; the identification of x depending on R is the order i− tuple and is written as If the different vertices of Γ have different codes based on R, then R is known as a resolving set for Γ. Let Ω ⊂ Γ of graph Γ be such that |Ω| � min |R|: R is a resolving set for Γ , then Ω is a metric basis for the graph Γ and dim(Γ) � |Ω| is metric dimension of the graph Γ.
Recently, the idea of edge metric dimension of graph (edim(Γ)) 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 P n , edim(P n ) � dim(P n ) � 1 where n is number of vertices of graph, cycle graph C n : If the different edges of Γ have different codes based on A, then A is known as an edge resolving set for Γ. Let L be a subset of vertex set of graph Γ such that |L| � min |L|: L is an edge resolving set for Γ , then L is an edge metric basis for the graph Γ and edim(Γ) � |L| is edge metric dimension of the graph Γ.
Kelenc et al. [20] has presented the idea of mixed metric dimension and denoted it as m di m(Γ), and they presented the mixed metric dimension of path graph P n , mdim(P n ) � 2, cycle graph C n , mdim(C n ) � 3, and complete bipartite graph K (r,t) , mdim(K (r,t) ) � r + t − 1 where r, t � 2; otherwise, mdim(K (r,t) ) � r + t − 2 and grid graph Γ � P r □P t , mdim(Γ) � 3 where r ≥ t ≥ 2. 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 Γ be a graph and X � x 1 , x 2 , . . . , x k be an ordered subset of vertices of graph Γ. If all vertices and edges of Γ have different codes of representation with respect to the set X, then X is known as a mixed resolving set for graph Γ. Let M be a subset of vertex set of graph Γ such that |M| � min : X is a mixed resolving set for graph Γ . en, M is known as a mixed metric basis for Γ, and mdim(Γ) � |M| is a mixed metric dimension of the graph Γ. In 2016, Yero [21] presented some bounds or closed formulae for the (edge, mixed) metric dimension of several families of graphs. e representation of all edges with respect to A is as given in Table 1.
All edges have different representations with respect to A, so edim(Γ) � 2.
e set B � v 2 , v 7 , v 8 is a mixed resolving set of Γ. e representation of all edges and vertices with respect to B is as given in Table 2.
All edges and vertices have different representations with respect to B, so mdim(Γ) � 3.
Theorem 1 (see [12]). A connected graph Γ of order n has dimension 1 if and only if Γ � P n .

edge metric dimension of a graph G is 1 if and only if Γ is a path.
Theorem 4 (see [20]). e mixed metric dimension of a graph G is 2 if and only if Γ is a path.

Results on Dragon Graph T n,m
Dragon graph is obtained by identifying vertex v n of cycle graph C n with vertex u 0 of path graph P m+1 and is denoted as Figure 2. Proof. Since dragon graph is not a path graph, edim(T n,m ) ≥ 2. In this case, A � v 1 , u m is an edge resolving set of T n,m . All edges are labeled as 2 Complexity e representations of all edges with respect to A are as follows: All edges have distinguished representation, and this fact can easily be verified, which implies that edim(T n,m ) ≤ 2. So, we obtained edim(T n,m ) � 2. Theorem 6. Let T n,m be a dragon graph for n ≥ 3 and m ≥ 2.
Proof. Since dragon graph is not a path graph, mdim(T n,m ) ≥ 3. All edges are labeled as In this case, A � v 1 , v ⌈n/2⌉ , u m is mixed resolving set of T n,m . e representations of all vertices with respect to A are as follows: For n even, For n odd, e representations of all edges with respect to A are as follows: For n even, c d e f g h Figure 1: A graph with the edim(Γ) � 2.
For n odd, From the above representation, it is clear that no two vertices, edges, and an edge or a vertex of T n,m have the same representation which implies that mdim(T n,m ) ≤ 3. So, we obtained mdim(T n,m ) � 3.

Results on Line Graph of Dragon Graph L(T n,m )
A line graph L(Γ) of Γ can be generated by assuming edge set of Γ as vertex set of L(Γ), and any two vertices of L(Γ) are adjacent if and only if they are neighboring edges in Γ. Line graph of dragon graph has V(L(T n,m )) � a 1 , a 2 , . . . , a n , a 1 b 1 , a n b 1 as its set of vertices and set of edges, where a n+1 � a 1 as shown in Figure 3. Proof. Since dim(L(T n,m )) ≥ 2, it is not a path graph P n . A � a 2 , b m is resolving set of L(T n,m ), and the representations of all vertices with respect to A are as follows: All vertices have different representation, which implies that dim(L(T n,m )) ≤ 2. So, we obtained dim(L(T n,m )) � 2. Proof. Since edim(L(T n,m )) ≥ 2, it is not a path graph P n . All edges of L(T n,m ) are labeled as e 1 ′ � a 1 b 1 and e 0 ′ � a n b 1 .
b m is an edge resolving set of L(T n,m ), and representations of all edges with respect to A are as follows: All edges have different representation, which implies that edim(L(T n,m )) ≤ 2. So, we obtained edim(L(T n,m )) � 2.

Complexity
Proof. Since mdim(L(T n,m )) ≥ 3, it is not a path graph P n . All edges of L(T n,m ) are labeled as e n � a 1 a n , A � a 2 , a ⌊n/2⌋+1 , b m is mixed resolving set of L(T n,m ), and the representations of all edges with respect to A are as follows: For n even, For n odd, e representations of all vertices with respect to A are as follows: From the above representation, it is clear that no two vertices, edges, and an edge or a vertex of L(T n,m ) have the same representation which implies that mdim(L(T n,m )) ≤ 3. So, we obtained mdim(L(T n,m )) � 3.

Results on Paraline Graph of Dragon Graph L(S(T n,m )))
Paraline graph is the line graph of the subdivision graph of any graph. In subdivision graph S � S(Γ), the vertex set is Proof. Since dim(L(S(T n,m ))) ≥ 2, it is not a path graph P n . A � a 2 , b 2m is resolving set of L(S(T n,m )), and the representations of all vertices with respect to A are All vertices have different representation, and this fact can easily be verified, which implies that dim(L(S(T n,m ))) ≤ 2. So, we obtained dim(L(S(T n,m ))) � 2. L(S(T n,m )) be a paraline graph of dragon graph for n ≥ 3 and m ≥ 2. en, edim(L(S(T n,m ))) � 2.

Complexity
Proof. Since edim(L(S(T n,m ))) ≥ 2, it is not a path graph P n . All edges of L(S(T n,m )) are labeled as e 1 ′ � a 1 b 1 and e 0 ′ � a 2 nb 1 .
A � a 2 , b 2m is an edge resolving set of L(S(T n,m )), and representations of all edges with respect to A are as follows: All edges have different representation, which implies that edim(L(S(T n,m ))) ≤ 2. So, we obtained edim(L(S (T n,m ))) � 2.
Proof. Since m di m(L(S(T n,m ))) ≥ 3, it is not a path graph P n . All edges of L(S(T n,m )) are labeled as e 2n � a 1 a 2n , A � a 2 , a n+1 , b 2m is mixed resolving set of L(S(T n,m )), and the representations of all edges with respect to A are as follows: e representations of all vertices with respect to A are as follows: (p − 2, n + 1 − p, 2m + p − 1) 2 ≤ p ≤ n; (p − 2, p − n − 1, 2(m + n) − p) p � n + 1; From the above representation, it is clear that no two vertices, edges, and an edge or a vertex of L(S(T n,m )) have the same representation which implies that mdim(L(S (T n,m ))) ≤ 3. So, we obtained mdim(L(S(T n,m ))) � 3.

Results on Line Graph of Dragon
Graph L(L(T n,m )) Let Γ be a graph and L(L(T n,m )) be line graph of line graph of dragon graph T n.m . Vertex set of line graph of line graph of dragon graph is VL((L(T n,m ))) � a 1 , a 2 , . . . , a n , b 0 , b 1 , b 2 , . . . , b m−1 }, and its edge set is E(L(T n,m )) � a i , a i+1 , b j b j+1 : 1 ≤ i ≤ n, 0 ≤ j ≤ m − 2} ∪ a n−1 b 0 , a n b 0 , a n b 1 , where a n+1 � a 1 as shown in Figure 5. Proof. Since dimL((L(T n,m ))) ≥ 2, it is not a path graph P n . A � a 2 , b m−1 is resolving set of L(L(T n,m )), and the representations of all vertices with respect to A are as follows: For n even, For n odd, All vertices have different representations, which imply that dim(L(L(T n,m ))) ≤ 2. So, we obtained dim(L(L (T n,m ))) � 2. Proof. Since edimL((L(T n,m ))) ≥ 2, it is not a path graph P n . All edges are labeled as e i � a i a i+1 : 1 ≤ i ≤ n and e 1 ″ � a n−1 b 0 , e 2 ″ � a n b 0 , e 3 ″ � a n b 1 , A � a 2 , b 0 , b 1 , b m−1 is an edge resolving set of L(L(T n,m )), and the representations of all edges with respect to A are as follows: For n even, For n odd, (2, 1, 1, m − 1) i � n − 1; (1, 1, 1, m − 1) i � n.
All edges have different representations, which implies that edim(L(L(T n,m ))) ≤ 4. On the other hand, we have to show that edim(L(L(T n,m ))) ≥ 4.
Suppose to the contrary that edim(L(L(T n,m ))) � 3, then we have the following possibilities. If the set A � a i , a j , a k where i, j, k � 1, 2, . . . , n and i ≠ j ≠ k is an edge resolving set for graph L(L(T n,m ), then some edges have same representations as shown in Table 3 and in Table 4.
If the set A � b s , b t , b u where s, t, u � 2, 3, . . . , m and s ≠ t ≠ u is an edge resolving set for graph L(L(T n,m ), then some edges have same representations as shown in Table 5.
If the set A � a n , b 0 , b 1 is an edge resolving set for graph L(L(T n,m ), then some edges have same representations as shown in Tables 6 and 7. e set A � a n , b 0 , b 1 is not an edge resolving set for graph L(L(T n,m ) because it did not resolve the following edges: Any combination of three vertices will not resolve the all edges of the graph L(L(T n,m ). Hence, there is no edge resolving set with three vertices for L(L(T n,m ) which implies that e di mL((L(T n,m ))) ≥ 4. So, we obtained edim(L(L (T n,m ))) � 4.
s t u e n s t u e n−1 s t u e n−2 s t u Table 6: Edge distance codes of L(L(T n,m )) w.r.t. A � a n , b 0 , b 1 .
Proof. Since edimL((L(T n,m ))) ≥ 3, it is not a path graph P n . All edges are labeled as e i � a i a i+1 : 1 ≤ i ≤ n and e j ′ � b j b j+1 : 0 ≤ j ≤ m − 2 . e 1 ″ � a n−1 b 0 , e 2 ″ � a n b 0 , e 3 ″ � a n b 1 , e 4 ″ � a 1 b 1 , If A � a 2 , a ⌈n/2⌉+1 , b 0 , b 1 , b m−1 is mixed resolving set of L(L(T n,m )), then the representation of all edges with respect to A is as follows: