The Vertex-Edge Resolvability of Some Wheel-Related Graphs

A vertex w ∈ V ( H ) distinguishes (or resolves) two elements (edges or vertices) a,z ∈ V ( H ) ∪ E ( H ) if d ( w,a ) ≠ d ( w,z ) . A set W m of vertices in a nontrivial connected graph H is said to be a mixed resolving set for H if every two diﬀerent elements (edges and vertices) of H are distinguished by at least one vertex of W m . The mixed resolving set with minimum cardinality in H is called the mixed metric dimension (vertex-edge resolvability) of H and denoted by m dim ( H ) . The aim of this research is to determine the mixed metric dimension of some wheel graph subdivisions. We speciﬁcally analyze and compare the mixed metric, edge metric, and metric dimensions of the graphs obtained after the wheel graphs’ spoke, cycle, and barycentric subdivisions. We also prove that the mixed resolving sets for some of these graphs are independent.


Introduction
Suppose H � (V, E) is a nontrivial, simple, and connected graph, where E represents a set of edges and V represents a set of vertices. e distance between two vertices a and w in an undirected graph H, denoted by d(a, w), is the length of a shortest a − w path in H. In [1], Kelenc et al. introduced the concept of mixed metric dimension in graphs.
is dimension of graph H is the mixture of metric and edge metric dimensions.
A vertex w ∈ V is said to resolve two vertices v 1 and v 2 in H if d(w, v 1 ) ≠ d(w, v 2 ). Let w be a vertex and W � v 1 , v 2 , v 3 , . . . , v p be an ordered subset of vertices in H. e metric coordinate (or metric representation) r(w|W) of w with respect to W is the p-tuple (d(w, v 1 ), d(w, v 2 ), d(w, v 3 ), . . . , d(w, v p )). en, W is said to be a resolving set (or metric generator) for H if for every pair of vertices v 1 , v 2 ∈ V with v 1 ≠ v 2 , we have r(v 1 |W) ≠ r(v 2 |W). A resolving set with minimum cardinality is called the metric basis of H, and the cardinality of the metric basis set is the metric dimension dim(H) of H.
Slater introduced the idea of metric dimension in [2], where the metric generators were referred to as locating sets due to some relation with the problem of uniquely recognizing the location of intruders in networks. Harary and Melter, on the contrary, independently proposed the same concept of the metric dimension of a graph in [3], where metric generators were referred to as resolving sets. Several works on the applications and theoretical properties of this invariant have also been published. Metric dimension has various significant applications in computer science, mathematics, social sciences, chemical sciences, etc. [4][5][6][7][8][9][10][11][12][13][14].
ere also exist some other variations of metric dimension in the literature: independent resolving sets [15], local metric dimension [16], solid metric dimension [11], fault-tolerant metric dimension [17], and so on.
e distance between an edge e � ax and a vertex w is defined as d(e, w) � d(ax, w) � min d(a, w), d(x, w) { }. A vertex w ∈ V is said to resolve two edges e 1 and e 2 in H if d(w, e 1 ) ≠ d(w, e 2 ). Let e be an edge and W E � v 1 , v 2 , v 3 , . . . , v p be an ordered subset of vertices in H. e edge metric codes r E (e|W E ) of e with respect to W E are the p-tuple (d(e, v 1 ), d(e, v 2 ), d(e, v 3 ), . . . , d(e, v p )). en, W E is said to be an edge resolving set for H if for every pair of edges e 1 , e 2 ∈ E with e 1 ≠ e 2 , we have r E (e 1 |W E ) ≠ r E (e 2 |W E ). An edge resolving set with minimum cardinality is called an edge metric basis for H, and the cardinality of this edge metric basis set is the edge metric dimension edim(H) of H.
For a connected graph H, we see that every vertex of H is uniquely recognized by a resolving set W of H, and every edge of H is uniquely recognized by an edge resolving set W E of H; the natural question is as follows: whether every resolving set W is also an edge resolving set W E for H and vice versa? Kelenc et al. in [18] proved that there exist some families of graphs for which the resolving set W is also an edge resolving set W E , but in general, this is not true for every graph H. Similarly, for every graph H, the edge resolving set is not necessarily a resolving set for H.
Let us define a set of elements as V ∪ E, i.e., each element is an edge or a vertex. A vertex w ∈ V is said to resolve two elements a and z from . en, W m is said to be a mixed resolving set for H if for every pair of distinct elements a 1 , a 2 ∈ V ∪ E, we have r m (a 1 |W m ) ≠ r m (a 2 |W m ). A mixed resolving set with minimum cardinality is called a mixed metric basis for H, and the cardinality of this mixed metric basis set is the mixed metric dimension mdim(H) of H. By the definition of the mixed metric dimension, it is clear that a mixed resolving set is both edge resolving set and a resolving set, so we have ere are several studies [1,19,20] related to the mixed metric dimension of various graphs, for instance, cycle graphs, antiprism graphs, prism graphs, and convex polytopes, but there are many graphs for which the mixed metric dimension has not been found yet, such as the graphs obtained by some subdivisions of the wheel graph W n,1 . So, in this paper, we will compute the mixed metric dimension of the graphs obtained after the barycentric, spoke, and cycle subdivisions of the wheel graph W n,1 .

Preliminaries
In this section, we give the definition of a wheel and its related graphs, as well as recall some existing results on the edge metric dimension, and the metric dimension of wheelrelated graphs.

Wheel Graph.
A vertex u in an undirected graph G is said to be the universal vertex if it is adjacent to all other vertices of G. A wheel graph W n,1 (n ≥ 3) is a graph with n + 1 vertices obtained by joining a single universal vertex to all of the vertices of a cycle graph C n . W n,1 has a vertex set V � v, k 1 , k 2 , k 3 , . . . , k n and an edge set E � vk j , k j k j+1 |1 ≤ j ≤ n}, where all of the indices are taken to be modulo n.
e edges k j k j+1 are called the cycle edges of W n,1 , and the edges vk j are called as the spokes of the wheel graph.
We state that a family 5 of nontrivial connected graphs has bounded mixed metric dimension if there exists a constant L > 0 for every graph H of 5 such that mdim(H) ≤ L; otherwise, 5 has an unbounded mixed metric dimension. If all of the graphs in 5 have the same mixed metric dimension, then 5 is referred to as a family with a constant mixed metric dimension. Cycles C n and paths P n for n ≥ 3 are the graph families with a constant mixed metric dimension.

Independent Mixed Resolving Set.
A set W m of vertices from H is said to be an independent mixed resolving set for H if W m is an independent as well as mixed resolving set.
Let WSS n,1 , WCS n,1 , and WBS n,1 be the graphs obtained from the wheel graph W n,1 after spoke, cycle, and barycentric subdivisions of W n,1 , respectively. Recently, the metric and edge metric dimension for these three wheel-related graphs have been computed, and in [21], Raza and Bataineh made a comparison between the metric dimension and the edge metric dimension for these wheel-related graphs. e edge metric dimension and the metric dimension for these three graphs are as follows.
Proposition 4 (see [23,22] is article is organized as follows: in Section 3, we will study the mixed metric dimension of the spoke subdivision of the wheel graph WSS n,1 . In Sections 4 and 5, we will study the mixed metric dimension of the cycle and barycentric subdivision of the wheel graph, i.e., WCS n,1 and WBS n,1 , respectively. We also give the comparative analysis for the mixed metric, edge metric, and metric dimension of the graphs obtained after the spoke, cycle, and barycentric subdivisions of the wheel graph. In Section 6, we conclude the obtained results.

Mixed Metric Dimension of the Spoke
Subdivision of W n,1 In this section, we determine the mixed metric dimension of the spoke subdivision of a wheel graph. is subdivided with a new vertex l j . e resulting graph so obtained is known as the spoke subdivision wheel graph (SSWG) and is denoted by WSS n,1 . SSWG has 3n edges, E(W n,1 ) � vl j , l j k j , k j k j+1 |1 ≤ j ≤ n , and 2n + 1 vertices, where all indices are taken to be modulo n (see Figure 1). In this section, we obtain the mixed metric dimension of SSWG WSS n,1 .
Proof. To prove that mdim(WSS n,1 ) ≤ n, we construct a mixed resolving set for WSS n,1 . Suppose W m � k 1 , k 2 , k 3 , . . . , k n ⊆V(WSS n,1 ) having n cycle vertices from WSS n,1 . We claim that W m is a mixed resolving set for WSS n,1 . Now, we can give mixed codes to each of the vertex and edge of WSS n,1 with respect to W m . e sets of mixed metric codes for the vertices v, l j , k j |1 ≤ j ≤ n of WSS n,1 are as follows: Next, the sets of mixed metric codes for the edges vl j , l j k j , k j k j+1 |1 ≤ j ≤ n of WSS n,1 are as follows: From these sets of mixed codes for WSS n,1 , we obtain that |A| � 1, |B| � |C| � |D| � |E| � |F| � n, and A ∩ B ∩ C ∩ D ∩ E ∩ F � ∅, implying W m to be a mixed resolving set for WSS n,1 , i.e., mdim(WSS n,1 ) ≤ n. Conversely, suppose, on the contrary, that there exists a mixed resolving set W m ⊆WSS n,1 such that |W m | < n. en, we have the following cases to be considered: In this case, we further have two subcases: . . , k n , then there exists at least one vertex k j such that k j ∉ W m . en, for an edge vl j and the vertex v, we have r m (vl j |W m ) � r m (v|W m ) � (2, 2, 2, . . . , 2), a contradiction. erefore, the set W m is not a mixed resolving set for WSS n, 1 .
. . , k n , then at least one vertex l i belongs to the set W m . en, there exists one k j ∉ W m , and the corresponding vertex l j ∉ W m . en, for an edge vl j and the vertex v, we have

Journal of Mathematics
, a contradiction. erefore, again, in this case, the set W m is not a mixed resolving set for WSS n, 1 .
In this case, we have two subcases: en, clearly, for an edge vl j and the vertex v, we have r m (vl j |W m ) � r m (v|W m ), a contradiction. erefore, the set W m is not a mixed resolving set for WSS n,1 . Subcase (ii): if at least one l j must belong to the set W m , then there exists at least one vertex k j ∉ W m , and the corresponding vertex l j ∉ W m . en, for an edge vl j and a vertex v, we have r m (vl j |W m ) � r m (v|W m ), a contradiction. erefore, again, in this case, the set W m is not a mixed resolving set for WSS n,1 . us, in all the cases, we have |W m | ≥ n, implying mdim(WSS n,1 ) � n, which completes the proof of the theorem. □ □ Remark 1. For the spoke subdivision wheel graph H � WSS n,1 , we find that dim(WSS n,1 ) < edim(WSS n,1 ) < mdim(WSS n,1 ) (using Propositions 1 and 3 and eorem 1). e comparison between these three dimensions of WSS n,1 is clearly shown in Figure 2, and the value of each dimension depends on the number of vertices n in WSS n,1 .

Mixed Metric Dimension of the Cycle Subdivision of W n,1
In this section, we determine the mixed metric dimension of the cycle subdivision of a wheel graph.
is subdivided with a new vertex l j . e resulting graph so obtained is known as the cycle subdivision wheel graph (CSWG) and is denoted by WCS n,1 . CSWG has 3n edges, E(WCS n,1 ) � vk j , k j l j , l j k j+1 |1 ≤ j ≤ n , and 2n + 1 vertices, where all indices are taken to be modulo n (see Figure 3). In this section, we obtain the mixed metric dimension of CSWG WCS n,1 .
Proof. To prove this, we first generate the mixed resolving sets for all the cases, obtaining the upper bounds depending on the positive integer n. en, in the end, we show that the lower bound (or reverse inequality) is the same as the upper bound to conclude the theorem.
Case (I): n ≡ 0(mod6). In this case, we have n � 6h, where h ≥ 2 and h ∈ N. Suppose an ordered subset W m � l 1 , l 2 , l 4 , l 5 , . . . , l n−2 , l n−1 � l 3i+1 , l 3i+2 |0 ≤ i ≤ 2h −1} of vertices in WCS n,1 with |W m | � 4h. Next, we claim that W m is the mixed resolving set for WCS n,1 . Now, we can give mixed codes to every vertex and edge of WCS n,1 with respect to W m . e sets of mixed metric codes for the vertices u � v, l j , k j |1 ≤ j ≤ n of WCS n,1 are as follows:  Journal of Mathematics Next, the sets of mixed metric codes for the edges vk j , k j l j , l j k j+1 |1 ≤ j ≤ n of WCS n,1 are as follows:
Next, we claim that W m is the mixed resolving set for WCS n,1 . Now, we can give mixed codes to every vertex and edge of WCS n,1 with respect to W m . e sets of mixed metric codes for the vertices u � v, l j , k j |1 ≤ j ≤ n of WCS n,1 are as follows: Journal of Mathematics 9 Next, the sets of mixed metric codes for the edges vk j , k j l j , l j k j+1 |1 ≤ j ≤ n of WCS n,1 are as follows: From these sets of mixed codes for WCS n,1 , we obtain that |A| � 1, |B| � |C| � |D| � |E| � |F| � n, and implying W m to be a mixed resolving set for WCS n,1 , i.e., mdim(WCS n,1 ) ≤ 4h + 2. Next, using equation (1) and Proposition 2, we find that mdim(WCS n,1 ) � 4h + 2, in this case. Case (V): n ≡ 4(mod6). In this case, we have n � 6h + 4, where h ≥ 2 and h ∈ N. Suppose an ordered subset W m � l 1 , l 2 , l 4 , l 5 , . . . , l n−3 , l n−2 , l n � l 3i+1 , l 3i+2 | 0 ≤ i ≤ 2h} ∪ l n of vertices in WCS n,1 with |W m | � 4h + 3. Next, we claim that W m is the mixed resolving set for WCS n,1 . Now, we can give mixed codes to every vertex and edge of WCS n,1 with respect to W m . e sets of mixed metric codes for the vertices u � v, l j , k j |1 ≤ j ≤ n of WCS n,1 are as follows: Next, the sets of mixed metric codes for the edges vk j , k j l j , l j k j+1 |1 ≤ j ≤ n of WCS n,1 are as follows:

Journal of Mathematics
From these sets of mixed codes for WCS n,1 , we obtain that |A| � 1, |B| � |C| � |D| � |E| � |F| � n, and A ∩ B ∩ C ∩ D ∩ E ∩ F � ∅, implying W m to be a mixed resolving set for WCS n,1 , i.e., mdim(WCS n,1 ) ≤ 4h + 3. Case (VI): n ≡ 5(mod6). In this case, we have n � 6h + 5, where h ≥ 1 and h ∈ N. Suppose an ordered subset W m � l 1 , l 2 , l 4 , l 5 , . . . , l n−1 , l n � l 3i+1 , l 3i+2 | 0 ≤ i ≤ 2h + 1} of vertices in WCS n,1 with |W m | � 4h + 4. Next, we claim that W m is the mixed resolving set for WCS n,1 . Now, we can give mixed codes to every vertex and edge of WCS n,1 with respect to W m . e sets of mixed metric codes for the vertices u � v, l j , k j |1 ≤ j ≤ n of WCS n,1 are as follows: Next, the sets of mixed metric codes for the edges vk j , k j l j , l j k j+1 |1 ≤ j ≤ n of WCS n,1 are as follows: From these sets of mixed codes for WCS n,1 , we obtain that |A| � 1, |B| � |C| � |D| � |E| � |F| � n, and A ∩ B ∩ C ∩ D ∩ E ∩ F � ∅, implying W m to be a mixed resolving set for WCS n,1 , i.e., mdim(WCS n,1 ) ≤ 4h + 4. Now, for the second, third, fifth, and sixth case, we obtain their lower bounds as follows.
For the second case, suppose that W m ⊂ V(WCS n,1 ) with |W m | < 4h + 1 is a mixed resolving set for WCS n,1 .
We have the following two cases to be considered: Subcase (i): if W m ⊈ k 1 , k 2 , k 3 , . . . , k n , then there must exist a vertex l j such that l j ∈ W m . en, there exists at least one vertex l i ∈ W m such that k i−1 , k i+1 ∉ W m . en, for the corresponding edges vk i−1 and vk i+1 , we have r m (vk i+1 |W m ) � r m (vk i−1 |W m ), a contradiction. erefore, W m is not a mixed resolving set for WCS n,1 in this case.
Subcase (ii): if W m ⊂ k 1 , k 2 , k 3 , . . . , k n , then there exist at least two vertices k i and k j such that k i , k j ∉ W m . en, for the edges vk i and vk j , we have r m (vk i |W m ) � r m (vk j |W m ), a contradiction.
erefore, W m is not a mixed resolving set for WCS n,1 in this case as well. us, |W m | ≥ 4h + 1. is completes the proof for the second case.
For rest of the cases, the pattern is the same as that in Case (II).

Mixed Metric Dimension of the Barycentric
Subdivision of W n,1