# The Vertex-Edge Resolvability of Some Wheel-Related Graphs

Authors: Bao-Hua Xing; Sunny Kumar Sharma; Vijay Kumar Bhat; Hassan Raza; Jia-Bao Liu
Journal: Journal of Mathematics
 (2021)

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## Abstract

A vertexw∈VH distinguishes (or resolves) two elements (edges or vertices) a,z∈VH∪EH if dw,a≠dw,z. A set Wm 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 Wm. The mixed resolving set with minimum cardinality in H is called the mixed metric dimension (vertex-edge resolvability) of H and denoted by mdimH. 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.


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## Body

## 1. Introduction

SupposeH=V,E is a nontrivial, simple, and connected graph, where E represents a set of edges and V represents a set of vertices. The distance between two vertices a and w in an undirected graph H, denoted by da,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. This dimension of graph H is the mixture of metric and edge metric dimensions.A vertexw∈V is said to resolve two vertices v1 and v2 in H if dw,v1≠dw,v2. Let w be a vertex and W=v1,v2,v3,…,vp be an ordered subset of vertices in H. The metric coordinate (or metric representation) rw|W of w with respect to W is the p-tuple dw,v1,dw,v2,dw,v3,…,dw,vp. Then, W is said to be a resolving set (or metric generator) for H if for every pair of vertices v1,v2∈V with v1≠v2, we have rv1|W≠rv2|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 dimH 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. [4–14]. There 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.The distance between an edgee=ax and a vertex w is defined as de,w=dax,w=minda,w,dx,w. A vertex w∈V is said to resolve two edges e1 and e2 in H if dw,e1≠dw,e2. Let e be an edge and WE=v1,v2,v3,…,vp be an ordered subset of vertices in H. The edge metric codes rEe|WE of e with respect to WE are the p-tuple de,v1,de,v2,de,v3,…,de,vp. Then, WE is said to be an edge resolving set for H if for every pair of edges e1,e2∈E with e1≠e2, we have rEe1|WE≠rEe2|WE. 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 edimH of H.For a connected graphH, 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 WE of H; the natural question is as follows: whether every resolving set W is also an edge resolving set WE 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 WE, 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 asV∪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 V∪E if dw,a≠dw,z. Let a be an element and Wm=v1,v2,v3,…,vp be an ordered subset of vertices in H. The mixed metric codes rma|Wm of a with respect to Wm are the p-tuple da,v1,da,v2,da,v3,…,da,vp. Then, Wm is said to be a mixed resolving set for H if for every pair of distinct elements a1,a2∈V∪E, we have rma1|Wm≠rma2|Wm. 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 mdimH 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(1)mdimH≥maxedimH,dimH.There 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 Wn,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 Wn,1.


## 2. 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 wheel-related graphs.


### 2.1. Wheel Graph

A vertexu 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 Wn,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 Cn. Wn,1 has a vertex set V=v,k1,k2,k3,…,kn and an edge set E=vkj,kjkj+1|1≤j≤n, where all of the indices are taken to be modulo n. The edges kjkj+1 are called the cycle edges of Wn,1, and the edges vkj are called as the spokes of the wheel graph.We state that a familyϜ of nontrivial connected graphs has bounded mixed metric dimension if there exists a constant L>0 for every graph H of Ϝ such that mdimH≤L; otherwise, Ϝ has an unbounded mixed metric dimension. If all of the graphs in Ϝ have the same mixed metric dimension, then Ϝ is referred to as a family with a constant mixed metric dimension. Cycles Cn and paths Pn for n≥3 are the graph families with a constant mixed metric dimension.


### 2.2. Independent Mixed Resolving Set

A setWm of vertices from H is said to be an independent mixed resolving set for H if Wm is an independent as well as mixed resolving set.LetWSSn,1, WCSn,1, and WBSn,1 be the graphs obtained from the wheel graph Wn,1 after spoke, cycle, and barycentric subdivisions of Wn,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. The edge metric dimension and the metric dimension for these three graphs are as follows.Proposition 1 (see [21]).
edimWSSn,1=n−1, for n≥6.Proposition 2 (see [21]).
Forn≥6, we have(2)edimWCSn,1=edimWBSn,1=4hifn=6horn=6h+1,4h+1ifn=6h+2,4h+2ifn=6h+3orn=6h+4,4h+3ifn=6h+5.Proposition 3 (see [22]).
dimWSSn,1=⌊2n+2/5⌋, for n≥6.Proposition 4 (see [23, 22]).
Forn≥6, we have(3)dimWCSn,1=dimWBSn,1=4hifn=6horn=6h+1,4h+1ifn=6h+2,4h+2ifn=6h+3orn=6h+4,4h+3ifn=6h+5.This article is organized as follows: in Section3, we will study the mixed metric dimension of the spoke subdivision of the wheel graph WSSn,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., WCSn,1 and WBSn,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.


## 2.1. Wheel Graph

A vertexu 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 Wn,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 Cn. Wn,1 has a vertex set V=v,k1,k2,k3,…,kn and an edge set E=vkj,kjkj+1|1≤j≤n, where all of the indices are taken to be modulo n. The edges kjkj+1 are called the cycle edges of Wn,1, and the edges vkj are called as the spokes of the wheel graph.We state that a familyϜ of nontrivial connected graphs has bounded mixed metric dimension if there exists a constant L>0 for every graph H of Ϝ such that mdimH≤L; otherwise, Ϝ has an unbounded mixed metric dimension. If all of the graphs in Ϝ have the same mixed metric dimension, then Ϝ is referred to as a family with a constant mixed metric dimension. Cycles Cn and paths Pn for n≥3 are the graph families with a constant mixed metric dimension.


## 2.2. Independent Mixed Resolving Set

A setWm of vertices from H is said to be an independent mixed resolving set for H if Wm is an independent as well as mixed resolving set.LetWSSn,1, WCSn,1, and WBSn,1 be the graphs obtained from the wheel graph Wn,1 after spoke, cycle, and barycentric subdivisions of Wn,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. The edge metric dimension and the metric dimension for these three graphs are as follows.Proposition 1 (see [21]).
edimWSSn,1=n−1, for n≥6.Proposition 2 (see [21]).
Forn≥6, we have(2)edimWCSn,1=edimWBSn,1=4hifn=6horn=6h+1,4h+1ifn=6h+2,4h+2ifn=6h+3orn=6h+4,4h+3ifn=6h+5.Proposition 3 (see [22]).
dimWSSn,1=⌊2n+2/5⌋, for n≥6.Proposition 4 (see [23, 22]).
Forn≥6, we have(3)dimWCSn,1=dimWBSn,1=4hifn=6horn=6h+1,4h+1ifn=6h+2,4h+2ifn=6h+3orn=6h+4,4h+3ifn=6h+5.This article is organized as follows: in Section3, we will study the mixed metric dimension of the spoke subdivision of the wheel graph WSSn,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., WCSn,1 and WBSn,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.


## 3. Mixed Metric Dimension of the Spoke Subdivision ofWn,1

In this section, we determine the mixed metric dimension of the spoke subdivision of a wheel graph.


### 3.1. Spoke Subdivision ofWn,1

SupposeWn,1 is a wheel graph with the vertex set VWn,1=k1,k2,k3,…,kn,v having a single universal vertex v. Now, each central spoke vkj of Wn,1 is subdivided with a new vertex lj. The resulting graph so obtained is known as the spoke subdivision wheel graph (SSWG) and is denoted by WSSn,1. SSWG has 3n edges, EWn,1=vlj,ljkj,kjkj+1|1≤j≤n, and 2n+1 vertices, VWn,1=v,lj,kj|1≤j≤n, where all indices are taken to be modulo n (see Figure 1). In this section, we obtain the mixed metric dimension of SSWG WSSn,1.Figure 1
WSSn,1.Theorem 1.
mdimWSSn,1=n, for n≥6.Proof.
To prove thatmdimWSSn,1≤n, we construct a mixed resolving set for WSSn,1. Suppose Wm=k1,k2,k3,…,kn⊆VWSSn,1 having n cycle vertices from WSSn,1. We claim that Wm is a mixed resolving set for WSSn,1. Now, we can give mixed codes to each of the vertex and edge of WSSn,1 with respect to Wm.
The sets of mixed metric codes for the verticesv,lj,kj|1≤j≤n of WSSn,1 are as follows:(4)A=rmv|Wm=2,2,2,…,2︸n−times,B=rmlj|Wm=3,3,…,3,2,1︸jth,2,3,…,3,3|1≤j≤n,C=rmkj|Wm=4,4,…,4,3,2,1,0︸jth,1,2,3,4,…,4,4|1≤j≤n.
Next, the sets of mixed metric codes for the edgesvlj,ljkj,kjkj+1|1≤j≤n of WSSn,1 are as follows:(5)D=rmvlj|Wm=2,2,…,2,1︸jth,2,…,2,2|1≤j≤n,E=rmljkj|Wm=3,3,…,3,2,1,0︸jth,1,2,3,…,3,3|1≤j≤n,F=rmkjkj+1|Wm=4,4,…,4,3,2,1,0︸jth,0,1,2,3,4,…,4,4|1≤j≤n.
From these sets of mixed codes forWSSn,1, we obtain that A=1, B=C=D=E=F=n, and A∩B∩C∩D∩E∩F=∅, implying Wm to be a mixed resolving set for WSSn,1, i.e., mdimWSSn,1≤n. Conversely, suppose, on the contrary, that there exists a mixed resolving set Wm⊆WSSn,1 such that Wm<n. Then, we have the following cases to be considered:Case (i):v∉Wm. In this case, we further have two subcases:Subcase (i): ifWm⊂k1,k2,k3,…,kn, then there exists at least one vertex kj such that kj∉Wm. Then, for an edge vlj and the vertex v, we have rmvlj|Wm=rmv|Wm=2,2,2,…,2, a contradiction. Therefore, the set Wm is not a mixed resolving set for WSSn,1.Subcase (ii): ifWm⊈k1,k2,k3,…,kn, then at least one vertex li belongs to the set Wm. Then, there exists one kj∉Wm, and the corresponding vertex lj∉Wm. Then, for an edge vlj and the vertex v, we have rmvlj|Wm=rmv|Wm, a contradiction. Therefore, again, in this case, the set Wm is not a mixed resolving set for WSSn,1.Case (ii):v∈Wm. In this case, we have two subcases:Subcase (i): ifWm⊂k1,k2,k3,…,kn∪v, then there exists at least one vertex kj such that kj∉Wm. Then, clearly, for an edge vlj and the vertex v, we have rmvlj|Wm=rmv|Wm, a contradiction. Therefore, the set Wm is not a mixed resolving set for WSSn,1.Subcase (ii): if at least onelj must belong to the set Wm, then there exists at least one vertex kj∉Wm, and the corresponding vertex lj∉Wm. Then, for an edge vlj and a vertex v, we have rmvlj|Wm=rmv|Wm, a contradiction. Therefore, again, in this case, the set Wm is not a mixed resolving set for WSSn,1. Thus, in all the cases, we have Wm≥n, implying mdimWSSn,1=n, which completes the proof of the theorem. □Remark 1.
For the spoke subdivision wheel graphH=WSSn,1, we find that dimWSSn,1<edimWSSn,1<mdimWSSn,1 (using Propositions 1 and 3 and Theorem 1). The comparison between these three dimensions of WSSn,1 is clearly shown in Figure 2, and the value of each dimension depends on the number of vertices n in WSSn,1.Figure 2
Comparison betweendimH, edimH, and mdimH.


## 3.1. Spoke Subdivision ofWn,1

SupposeWn,1 is a wheel graph with the vertex set VWn,1=k1,k2,k3,…,kn,v having a single universal vertex v. Now, each central spoke vkj of Wn,1 is subdivided with a new vertex lj. The resulting graph so obtained is known as the spoke subdivision wheel graph (SSWG) and is denoted by WSSn,1. SSWG has 3n edges, EWn,1=vlj,ljkj,kjkj+1|1≤j≤n, and 2n+1 vertices, VWn,1=v,lj,kj|1≤j≤n, where all indices are taken to be modulo n (see Figure 1). In this section, we obtain the mixed metric dimension of SSWG WSSn,1.Figure 1
WSSn,1.Theorem 1.
mdimWSSn,1=n, for n≥6.Proof.
To prove thatmdimWSSn,1≤n, we construct a mixed resolving set for WSSn,1. Suppose Wm=k1,k2,k3,…,kn⊆VWSSn,1 having n cycle vertices from WSSn,1. We claim that Wm is a mixed resolving set for WSSn,1. Now, we can give mixed codes to each of the vertex and edge of WSSn,1 with respect to Wm.
The sets of mixed metric codes for the verticesv,lj,kj|1≤j≤n of WSSn,1 are as follows:(4)A=rmv|Wm=2,2,2,…,2︸n−times,B=rmlj|Wm=3,3,…,3,2,1︸jth,2,3,…,3,3|1≤j≤n,C=rmkj|Wm=4,4,…,4,3,2,1,0︸jth,1,2,3,4,…,4,4|1≤j≤n.
Next, the sets of mixed metric codes for the edgesvlj,ljkj,kjkj+1|1≤j≤n of WSSn,1 are as follows:(5)D=rmvlj|Wm=2,2,…,2,1︸jth,2,…,2,2|1≤j≤n,E=rmljkj|Wm=3,3,…,3,2,1,0︸jth,1,2,3,…,3,3|1≤j≤n,F=rmkjkj+1|Wm=4,4,…,4,3,2,1,0︸jth,0,1,2,3,4,…,4,4|1≤j≤n.
From these sets of mixed codes forWSSn,1, we obtain that A=1, B=C=D=E=F=n, and A∩B∩C∩D∩E∩F=∅, implying Wm to be a mixed resolving set for WSSn,1, i.e., mdimWSSn,1≤n. Conversely, suppose, on the contrary, that there exists a mixed resolving set Wm⊆WSSn,1 such that Wm<n. Then, we have the following cases to be considered:Case (i):v∉Wm. In this case, we further have two subcases:Subcase (i): ifWm⊂k1,k2,k3,…,kn, then there exists at least one vertex kj such that kj∉Wm. Then, for an edge vlj and the vertex v, we have rmvlj|Wm=rmv|Wm=2,2,2,…,2, a contradiction. Therefore, the set Wm is not a mixed resolving set for WSSn,1.Subcase (ii): ifWm⊈k1,k2,k3,…,kn, then at least one vertex li belongs to the set Wm. Then, there exists one kj∉Wm, and the corresponding vertex lj∉Wm. Then, for an edge vlj and the vertex v, we have rmvlj|Wm=rmv|Wm, a contradiction. Therefore, again, in this case, the set Wm is not a mixed resolving set for WSSn,1.Case (ii):v∈Wm. In this case, we have two subcases:Subcase (i): ifWm⊂k1,k2,k3,…,kn∪v, then there exists at least one vertex kj such that kj∉Wm. Then, clearly, for an edge vlj and the vertex v, we have rmvlj|Wm=rmv|Wm, a contradiction. Therefore, the set Wm is not a mixed resolving set for WSSn,1.Subcase (ii): if at least onelj must belong to the set Wm, then there exists at least one vertex kj∉Wm, and the corresponding vertex lj∉Wm. Then, for an edge vlj and a vertex v, we have rmvlj|Wm=rmv|Wm, a contradiction. Therefore, again, in this case, the set Wm is not a mixed resolving set for WSSn,1. Thus, in all the cases, we have Wm≥n, implying mdimWSSn,1=n, which completes the proof of the theorem. □Remark 1.
For the spoke subdivision wheel graphH=WSSn,1, we find that dimWSSn,1<edimWSSn,1<mdimWSSn,1 (using Propositions 1 and 3 and Theorem 1). The comparison between these three dimensions of WSSn,1 is clearly shown in Figure 2, and the value of each dimension depends on the number of vertices n in WSSn,1.Figure 2
Comparison betweendimH, edimH, and mdimH.


## 4. Mixed Metric Dimension of the Cycle Subdivision ofWn,1

In this section, we determine the mixed metric dimension of the cycle subdivision of a wheel graph.


### 4.1. Cycle Subdivision ofWn,1

SupposeWn,1 is a wheel graph with the vertex set VWn,1=k1,k2,k3,…,kn,v having a single universal vertex v. Now, each cycle edge kjkj+1 of Wn,1 is subdivided with a new vertex lj. The resulting graph so obtained is known as the cycle subdivision wheel graph (CSWG) and is denoted by WCSn,1. CSWG has 3n edges, EWCSn,1=vkj,kjlj,ljkj+1|1≤j≤n, and 2n+1 vertices, VWCSn,1=v,lj,kj|1≤j≤n, where all indices are taken to be modulo n (see Figure 3). In this section, we obtain the mixed metric dimension of CSWG WCSn,1.Figure 3
WCSn,1.Theorem 2.
Forn≥6, we have(6)mdimWCSn,1=4hifn=6h,4h+1ifn=6h+1,4h+2ifn=6h+2,4h+2ifn=6h+3,4h+3ifn=6h+4,4h+4ifn=6h+5.Proof.
To prove this, we first generate the mixed resolving sets for all the cases, obtaining the upper bounds depending on the positive integern. Then, 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≡0mod6. In this case, we have n=6h, where h≥2 and h∈ℕ. Suppose an ordered subset Wm=l1,l2,l4,l5,…,ln−2,ln−1=l3i+1,l3i+2|0≤i≤2h−1 of vertices in WCSn,1 with Wm=4h. Next, we claim that Wm is the mixed resolving set for WCSn,1. Now, we can give mixed codes to every vertex and edge of WCSn,1 with respect to Wm. The sets of mixed metric codes for the vertices u=v,lj,kj|1≤j≤n of WCSn,1 are as follows:(7)A=rmv|Wm=2,2,2,…,2︸4h−times,B=rmkj|Wm=3,3,3,…,3,dl3i+2,k3i+3=1,3,…,3|j≡0mod30≤i≤2h−1∪rmkj|Wm=3,3,3,…,3,dl3i+1,k3i+1=1,3,…,3|j≡1mod30≤i≤2h−1∪rmkj|Wm=3,3,3,…,3,dl3i+1,k3i+2=1,dl3i+2,k3i+2=1,3,…,3|j≡2mod30≤i≤2h−1,C=rmlj|Wm=4,,4,…,4,dl3i+2,l3i+3=2,dl3i+4,l3i+3=2,4,…,4|j≡0mod30≤i≤2h−1∪rmlj|Wm=4,4,4,…,4,dl3i+1,l3i+1=0,dl3i+1,l3i+2=2,4,…,4|j≡1mod30≤i≤2h−1∪rmlj|Wm=4,4,4,…,4,dl3i+1,l3i+1=2,dl3i+1,l3i+2=0,4,…,4|j≡2mod30≤i≤2h−1.Next, the sets of mixed metric codes for the edgesvkj,kjlj,ljkj+1|1≤j≤n of WCSn,1 are as follows:(8)D=rmvkj∣Wm=2,2,2,…,2,dl3i+2,vk3i+3=1,2,…,2∣j≡0mod3&0≤i≤2h−1∪rmvkj∣Wm=2,2,2,…,2,dl3i+1,vk3i+2=1,2,…,2∣j≡1mod3&0≤i≤2h−1∪rmvkj∣Wm=2,2,2,…,2,dl3i+1,vk3i+2=1,dl3i+2,vk3i+2=1,2,…,2∣j≡2mod3&0≤i≤2h−1;E=rmkjlj∣Wm=3,3,3,…,3,dl3i+2,k3i+3l3i+3=1,dl3i+4,k3i+3l3i+3=2,3,…,3∣j≡0mod3&0≤i≤2h−1∪rmkjlj∣Wm=3,3,3,…,3,dl3i+1,k3i+1l3i+1=0,dl3i+2,k3i+1l3i+1=2,3,…,3∣j≡1mod3&0≤i≤2h−1∪rmkjlj∣Wm=3,3,3,…,3,dl3i+1,k3i+2l3i+2=1,dl3i+2,k3i+2l3i+2=0,3,…,3∣j≡2mod3&0≤i≤2h−1;F=rmljkj+1∣Wm=3,3,3,…,3,dl3i+2,l3i+3k3i+4=2,dl3i+4,l3i+3k3i+4=1,3,…,3∣j≡0mod3&0≤i≤2h−1∪rmljkj+1∣Wm=3,3,3,…,3,dl3i+1,l3i+1k3i+2=0,dl3i+2,l3i+1k3i+2=1,3,…,3∣j≡1mod3&0≤i≤2h−1∪rmljkj+1∣Wm=3,3,3,…,3,dl3i+1,l3i+2k3i+3=2,dl3i+2,l3i+2k3i+3=0,3,…,3∣j≡2mod3&0≤i≤2h−1.From these sets of mixed codes forWCSn,1, we obtain that A=1, B=C=D=E=F=n, and A∩B∩C∩D∩E∩F=∅, implying Wm to be a mixed resolving set for WCSn,1, i.e., mdimWCSn,1≤4h. Next, using equation (1) and Proposition 2, we find that mdimWCSn,1=4h, in this case.Case (II):n≡1mod6. In this case, we have n=6h+1, where h≥2 and h∈ℕ. Suppose an ordered subset Wm=l1,l2,l4,l5,…,ln−3,ln−2,ln=l3i+1,l3i+2|0≤i≤2h−1∪ln of vertices in WCSn,1 with Wm=4h+1. Next, we claim that Wm is the mixed resolving set for WCSn,1. Now, we can give mixed codes to every vertex and edge of WCSn,1 with respect to Wm. The sets of mixed metric codes for the vertices u=v,lj,kj|1≤j≤n of WCSn,1 are as follows:(9)A=rmv|Wm=2,2,2,…,2︸4h+1−times;B=rmkj|Wm=3,3,3,…,3,dl3i+2,k3i+3=1,3,…,3|j≡0mod30≤i≤2h−1∪rmk1|Wm=1,3,3,…,3︸4h−1−times,1∪rmkj|Wm=3,3,3,…,3,dl3i+1,k3i+1=1,3,…,3|j≡1mod31≤i≤2h∪rmkj|Wm=3,3,3,…,3,dl3i+1,k3i+2=1,dl3i+2,k3i+2=1,3,…,3|j≡2mod30≤i≤2h−1,C=rmlj|Wm=4,,4,…,4,dl3i+2,l3i+3=2,dl3i+4,l3i+3=2,4,…,4|j≡0mod30≤i≤2h−1∪rml1|Wm=0,2,4,4,4,…,4︸4h−2−times,2∪rmlj|Wm=4,4,4,…,4,dl3i+1,l3i+1=0,dl3i+1,l3i+2=2,4,…,4|j≡1mod31≤i≤2h∪rmlj|Wm=4,4,4,…,4,dl3i+1,l3i+1=2,dl3i+1,l3i+2=0,4,…,4|j≡2mod30≤i≤2h−1.Next, the sets of mixed metric codes for the edgesvkj,kjlj,ljkj+1|1≤j≤n of WCSn,1 are as follows:(10)D=rmvkj|Wm=2,2,2,…,2,dl3i+2,vk3i+3=1,2,…,2|j≡0mod30≤i≤2h−1∪rmvk1|Wm=1,2,2,2,…,2︸4h−1−times,1∪rmvkj|Wm=2,2,2,…,2,dl3i+1,vk3i+1=1,2,…,2|j≡1mod31≤i≤2h∪rmvkj|Wm=2,2,2,…,2,dl3i+1,vk3i+2=1,dl3i+2,vk3i+2=1,2,…,2|j≡2mod30≤i≤2h−1,E=rmkjlj|Wm=3,3,3,…,3,dl3i+2,k3i+3l3i+3=1,dl3i+4,k3i+3l3i+3=2,3,…,3|j≡0mod30≤i≤2h−1∪rmk1l1|Wm=0,2,3,3,3,…,3︸4h−2−times,1∪rmkjlj|Wm=3,3,3,…,3,dl3i+1,k3i+1l3i+1=0,dl3i+2,k3i+1l3i+1=2,3,…,3|j≡1mod31≤i≤2h∪rmkjlj|Wm=3,3,3,…,3,dl3i+1,k3i+2l3i+2=1,dl3i+2,k3i+2l3i+2=0,3,…,3|j≡2mod30≤i≤2h−1,F=rmljkj+1|Wm=3,3,3,…,3,dl3i+2,l3i+3k3i+4=2,dl3i+4,l3i+3k3i+4=1,3,…,3|j≡0mod30≤i≤2h−1∪rml1k2|Wm=0,1,3,3,3,…,3︸4h−2−times,1∪rmljkj+1|Wm=3,3,3,…,3,dl3i+1,l3i+1k3i+2=0,dl3i+2,l3i+1k3i+2=1,3,…,3|j≡1mod31≤i≤2h∪rmljkj+1|Wm=3,3,3,…,3,dl3i+1,l3i+2k3i+3=2,dl3i+2,l3i+2k3i+3=0,3,…,3|j≡2mod30≤i≤2h−1.From these sets of mixed codes forWCSn,1, we obtain that A=1, B=C=D=E=F=n, and A∩B∩C∩D∩E∩F=∅, implying Wm to be a mixed resolving set for WCSn,1, i.e., mdimWCSn,1≤4h+1.Case (III):n≡2mod6. In this case, we have n=6h+2, where h≥2 and h∈ℕ. Suppose an ordered subset Wm=l1,l2,l4,l5,l7…,ln−1,ln=l3i+1,l3i+2|0≤i≤2h of vertices in WCSn,1 with Wm=4h+2. Next, we claim that Wm is the mixed resolving set for WCSn,1. Now, we can give mixed codes to every vertex and edge of WCSn,1 with respect to Wm. The sets of mixed metric codes for the vertices u=v,lj,kj|1≤j≤n of WCSn,1 are as follows:(11)A=rmv|Wm=2,2,2,…,2︸4h+2−times,B=rmkj|Wm=3,3,3,…,3,dl3i+2,k3i+3=1,3,…,3|j≡0mod30≤i≤2h−1∪rmk1|Wm=1,3,3,…,3︸4h−times,1∪rmkj|Wm=3,3,3,…,3,dl3i+1,k3i+1=1,3,…,3|j≡1mod31≤i≤2h∪rmkj|Wm=3,3,3,…,3,dl3i+1,k3i+2=1,dl3i+2,k3i+2=1,3,…,3|j≡2mod30≤i≤2h,C=rmlj|Wm=4,,4,…,4,dl3i+2,l3i+3=2,dl3i+4,l3i+3=2,4,…,4|j≡0mod30≤i≤2h−1∪rml1|Wm=0,2,4,4,4,…,4︸4h−1−times,2∪rmlj|Wm=4,4,4,…,4,dl3i+1,l3i+1=0,dl3i+1,l3i+2=2,4,…,4|j≡1mod31≤i≤2h∪rmlj|Wm=4,4,4,…,4,dl3i+1,l3i+2=2,dl3i+2,l3i+2=0,4,…,4|j≡2mod30≤i≤2h.Next, the sets of mixed metric codes for the edgesvkj,kjlj,ljkj+1|1≤j≤n of WCSn,1 are as follows:(12)D=rmvkj|Wm=2,2,2,…,2,dl3i+2,vk3i+3=1,2,…,2|j≡0mod30≤i≤2h−1∪rmvk1|Wm=1,2,2,2,…,2︸4h−times,1∪rmvkj|Wm=2,2,2,…,2,dl3i+1,vk3i+1=1,2,…,2|j≡1mod31≤i≤2h∪rmvkj|Wm=2,2,2,…,2,dl3i+1,vk3i+2=1,dl3i+2,vk3i+2=1,2,…,2|j≡2mod30≤i≤2h,E=rmkjlj|Wm=3,3,3,…,3,dl3i+2,k3i+3l3i+3=1,dl3i+4,k3i+3l3i+3=2,3,…,3|j≡0mod30≤i≤2h−1∪rmk1l1|Wm=0,2,3,3,3,…,3︸4h−1−times,1∪rmkjlj|Wm=3,3,3,…,3,dl3i+1,k3i+1l3i+1=0,dl3i+2,k3i+1l3i+1=2,3,…,3|j≡1mod31≤i≤2h∪rmkjlj|Wm=3,3,3,…,3,dl3i+1,k3i+2l3i+2=1,dl3i+2,k3i+2l3i+2=0,3,…,3|j≡2mod30≤i≤2h,F=rmljkj+1|Wm=3,3,3,…,3,dl3i+2,l3i+3k3i+4=2,dl3i+4,l3i+3k3i+4=1,3,…,3|j≡0mod30≤i≤2h−1∪rml1k2|Wm=0,1,3,3,3,…,3︸4h−1−times,1∪rmljkj+1|Wm=3,3,3,…,3,dl3i+1,l3i+1k3i+2=0,dl3i+2,l3i+1k3i+2=1,3,…,3|j≡1mod31≤i≤2h∪rmljkj+1|Wm=3,3,3,…,3,dl3i+1,l3i+2k3i+3=2,dl3i+2,l3i+2k3i+3=0,3,…,3|j≡2mod30≤i≤2h.From these sets of mixed codes forWCSn,1, we obtain that A=1, B=C=D=E=F=n, and A∩B∩C∩D∩E∩F=∅, implying Wm to be a mixed resolving set for WCSn,1, i.e., mdimWCSn,1≤4h+2.Case (IV):n≡3mod6. In this case, we have n=6h+3, where h≥2 and h∈ℕ. Suppose an ordered subset Wm=l1,l2,l4,l5,l7…,ln−2,ln−1=l3i+1,l3i+2|0≤i≤2h of vertices in WCSn,1 with Wm=4h+2. Next, we claim that Wm is the mixed resolving set for WCSn,1. Now, we can give mixed codes to every vertex and edge of WCSn,1 with respect to Wm. The sets of mixed metric codes for the vertices u=v,lj,kj|1≤j≤n of WCSn,1 are as follows:(13)A=rmv|Wm=2,2,2,…,2︸4h+2−times,B=rmkj|Wm=3,3,3,…,3,dl3i+2,k3i+3=1,3,…,3|j≡0mod30≤i≤2h∪rmkj|Wm=3,3,3,…,3,dl3i+1,k3i+1=1,3,…,3|j≡1mod30≤i≤2h∪rmkj|Wm=3,3,3,…,3,dl3i+1,k3i+2=1,dl3i+2,k3i+2=1,3,…,3|j≡2mod30≤i≤2h,C=rmlj|Wm=4,,4,…,4,dl3i+2,l3i+3=2,dl3i+4,l3i+3=2,4,…,4|j≡0mod30≤i≤2h∪rmlj|Wm=4,4,4,…,4,dl3i+1,l3i+1=0,dl3i+1,l3i+2=2,4,…,4|j≡1mod30≤i≤2h∪rmlj|Wm=4,4,4,…,4,dl3i+1,l3i+2=2,dl3i+2,l3i+2=0,4,…,4|j≡2mod30≤i≤2h.Next, the sets of mixed metric codes for the edgesvkj,kjlj,ljkj+1|1≤j≤n of WCSn,1 are as follows:(14)D=rmvkj∣Wm=2,2,2,…,2,dl3i+2,vk3i+3=1,2,…,2∣j≡0mod3&0≤i≤2h∪rmvkj∣Wm=2,2,2,…,2,dl3i+1,vk3i+2=1,2,…,2∣j≡1mod3&0≤i≤2h∪rmvkj∣Wm=2,2,2,…,2,dl3i+1,vk3i+2=1,dl3i+2,vk3i+2=1,2,…,2∣j≡2mod3&0≤i≤2h;E=rmkjlj∣Wm=3,3,3,…,3,dl3i+2,k3i+3l3i+3=1,dl3i+4,k3i+3l3i+3=2,3,…,3∣j≡0mod3&0≤i≤2h∪rmkjlj∣Wm=3,3,3,…,3,dl3i+1,k3i+1l3i+1=0,dl3i+2,k3i+1l3i+1=2,3,…,3∣j≡1mod3&0≤i≤2h∪rmkjlj∣Wm=3,3,3,…,3,dl3i+1,k3i+2l3i+2=1,dl3i+2,k3i+2l3i+2=0,3,…,3∣j≡2mod3&0≤i≤2h;F=rmljkj+1∣Wm=3,3,3,…,3,dl3i+2,l3i+3k3i+4=2,dl3i+4,l3i+3k3i+4=1,3,…,3∣j≡0mod3&0≤i≤2h∪rmljkj+1∣Wm=3,3,3,…,3,dl3i+1,l3i+1k3i+2=0,dl3i+2,l3i+1k3i+2=1,3,…,3∣j≡1mod3&0≤i≤2h.∪rmljkj+1∣Wm=3,3,3,…,3,dl3i+1,l3i+2k3i+3=2,dl3i+2,l3i+2k3i+3=0,3,…,3∣j≡2mod3&0≤i≤2h;From these sets of mixed codes forWCSn,1, we obtain that A=1, B=C=D=E=F=n, and A∩B∩C∩D∩E∩F=∅, implying Wm to be a mixed resolving set for WCSn,1, i.e., mdimWCSn,1≤4h+2. Next, using equation (1) and Proposition 2, we find that mdimWCSn,1=4h+2, in this case.Case (V):n≡4mod6. In this case, we have n=6h+4, where h≥2 and h∈ℕ. Suppose an ordered subset Wm=l1,l2,l4,l5,…,ln−3,ln−2,ln=l3i+1,l3i+2|0≤i≤2h∪ln of vertices in WCSn,1 with Wm=4h+3. Next, we claim that Wm is the mixed resolving set for WCSn,1. Now, we can give mixed codes to every vertex and edge of WCSn,1 with respect to Wm. The sets of mixed metric codes for the vertices u=v,lj,kj|1≤j≤n of WCSn,1 are as follows:(15)A=rmv|Wm=2,2,2,…,2︸4h+3−times,B=rmkj|Wm=3,3,3,…,3,dl3i+2,k3i+3=1,3,…,3|j≡0mod30≤i≤2h∪rmk1|Wm=1,3,3,…,3︸4h+1−times,1∪rmkj|Wm=3,3,3,…,3,dl3i+1,k3i+1=1,3,…,3|j≡1mod31≤i≤2h+1∪rmkj|Wm=3,3,3,…,3,dl3i+1,k3i+2=1,dl3i+2,k3i+2=1,3,…,3|j≡2mod30≤i≤2h+1,C=rmlj|Wm=4,,4,…,4,dl3i+2,l3i+3=2,dl3i+4,l3i+3=2,4,…,4|j≡0mod30≤i≤2h∪rml1|Wm=0,2,4,4,4,…,4︸4h−times,2∪rmlj|Wm=4,4,4,…,4,dl3i+1,l3i+1=0,dl3i+1,l3i+2=2,4,…,4|j≡1mod31≤i≤2h+1∪rmlj|Wm=4,4,4,…,4,dl3i+1,l3i+1=2,dl3i+1,l3i+2=0,4,…,4|j≡2mod30≤i≤2h+1.Next, the sets of mixed metric codes for the edgesvkj,kjlj,ljkj+1|1≤j≤n of WCSn,1 are as follows:(16)D=rmvkj|Wm=2,2,2,…,2,dl3i+2,vk3i+3=1,2,…,2|j≡0mod30≤i≤2h∪rmvk1|Wm=1,2,2,2,…,2︸4h+1−times,1∪rmvkj|Wm=2,2,2,…,2,dl3i+1,vk3i+1=1,2,…,2|j≡1mod31≤i≤2h+1∪rmvkj|Wm=2,2,2,…,2,dl3i+1,vk3i+2=1,dl3i+2,vk3i+2=1,2,…,2|j≡2mod30≤i≤2h,E=rmkjlj|Wm=3,3,3,…,3,dl3i+2,k3i+3l3i+3=1,dl3i+4,k3i+3l3i+3=2,3,…,3|j≡0mod30≤i≤2h∪rmk1l1|Wm=0,2,3,3,3,…,3︸4h−times,1∪rmkjlj|Wm=3,3,3,…,3,dl3i+1,k3i+1l3i+1=0,dl3i+2,k3i+1l3i+1=2,3,…,3|j≡1mod31≤i≤2h+1∪rmkjlj|Wm=3,3,3,…,3,dl3i+1,k3i+2l3i+2=1,dl3i+2,k3i+2l3i+2=0,3,…,3|j≡2mod30≤i≤2h+1,F=rmljkj+1|Wm=3,3,3,…,3,dl3i+2,l3i+3k3i+4=2,dl3i+4,l3i+3k3i+4=1,3,…,3|j≡0mod30≤i≤2h∪rml1k2|Wm=0,1,3,3,3,…,3︸4h−times,1∪rmljkj+1|Wm=3,3,3,…,3,dl3i+1,l3i+1k3i+2=0,dl3i+2,l3i+1k3i+2=1,3,…,3|j≡1mod31≤i≤2h+1∪rmljkj+1|Wm=3,3,3,…,3,dl3i+1,l3i+2k3i+3=2,dl3i+2,l3i+2k3i+3=0,3,…,3|j≡2mod30≤i≤2h.From these sets of mixed codes forWCSn,1, we obtain that A=1,B=C=D=E=F=n, and A∩B∩C∩D∩E∩F=∅, implying Wm to be a mixed resolving set for WCSn,1, i.e., mdimWCSn,1≤4h+3.Case (VI):n≡5mod6. In this case, we have n=6h+5, where h≥1 and h∈ℕ. Suppose an ordered subset Wm=l1,l2,l4,l5,…,ln−1,ln=l3i+1,l3i+2|0≤i≤2h+1 of vertices in WCSn,1 with Wm=4h+4. Next, we claim that Wm is the mixed resolving set for WCSn,1. Now, we can give mixed codes to every vertex and edge of WCSn,1 with respect to Wm. The sets of mixed metric codes for the vertices u=v,lj,kj|1≤j≤n of WCSn,1 are as follows:(17)A=rmv|Wm=2,2,2,…,2︸4h+4−times,B=rmkj|Wm=3,3,3,…,3,dl3i+2,k3i+3=1,3,…,3|j≡0mod30≤i≤2h∪rmk1|Wm=1,3,3,…,3︸4h+2−times,1∪rmkj|Wm=3,3,3,…,3,dl3i+1,k3i+1=1,3,…,3|j≡1mod31≤i≤2h+1∪rmkj|Wm=3,3,3,…,3,dl3i+1,k3i+2=1,dl3i+2,k3i+2=1,3,…,3|j≡2mod30≤i≤2h+1,C=rmlj|Wm=4,,4,…,4,dl3i+2,l3i+3=2,dl3i+4,l3i+3=2,4,…,4|j≡0mod30≤i≤2h∪rml1|Wm=0,2,4,4,4,…,4︸4h+1−times,2∪rmlj|Wm=4,4,4,…,4,dl3i+1,l3i+1=0,dl3i+1,l3i+2=2,4,…,4|j≡1mod31≤i≤2h+1∪rmlj|Wm=4,4,4,…,4,dl3i+1,l3i+2=2,dl3i+2,l3i+2=0,4,…,4|j≡2mod30≤i≤2h+1.Next, the sets of mixed metric codes for the edgesvkj,kjlj,ljkj+1|1≤j≤n of WCSn,1 are as follows:(18)D=rmvkj|Wm=2,2,2,…,2,dl3i+2,vk3i+3=1,2,…,2|j≡0mod30≤i≤2h∪rmvk1|Wm=1,2,2,2,…,2︸4h+2−times,1∪rmvkj|Wm=2,2,2,…,2,dl3i+1,vk3i+1=1,2,…,2|j≡1mod31≤i≤2h+1∪rmvkj|Wm=2,2,2,…,2,dl3i+1,vk3i+2=1,dl3i+2,vk3i+2=1,2,…,2|j≡2mod30≤i≤2h+1,E=rmkjlj|Wm=3,3,3,…,3,dl3i+2,k3i+3l3i+3=1,dl3i+4,k3i+3l3i+3=2,3,…,3|j≡0mod30≤i≤2h∪rmk1l1|Wm=0,2,3,3,3,…,3︸4h+1−times,1∪rmkjlj|Wm=3,3,3,…,3,dl3i+1,k3i+1l3i+1=0,dl3i+2,k3i+1l3i+1=2,3,…,3|j≡1mod31≤i≤2h+1∪rmkjlj|Wm=3,3,3,…,3,dl3i+1,k3i+2l3i+2=1,dl3i+2,k3i+2l3i+2=0,3,…,3|j≡2mod30≤i≤2h+1,F=rmljkj+1|Wm=3,3,3,…,3,dl3i+2,l3i+3k3i+4=2,dl3i+4,l3i+3k3i+4=1,3,…,3|j≡0mod30≤i≤2h∪rml1k2|Wm=0,1,3,3,3,…,3︸4h+1−times,2∪rmljkj+1|Wm=3,3,3,…,3,dl3i+1,l3i+1k3i+2=0,dl3i+2,l3i+1k3i+2=1,3,…,3|j≡1mod31≤i≤2h+1∪rmljkj+1|Wm=3,3,3,…,3,dl3i+1,l3i+2k3i+3=2,dl3i+2,l3i+2k3i+3=0,3,…,3|j≡2mod30≤i≤2h+1.From these sets of mixed codes forWCSn,1, we obtain that A=1, B=C=D=E=F=n, and A∩B∩C∩D∩E∩F=∅, implying Wm to be a mixed resolving set for WCSn,1, i.e., mdimWCSn,1≤4h+4. Now, for the second, third, fifth, and sixth case, we obtain their lower bounds as follows.For the second case, suppose thatWm⊂VWCSn,1 with Wm<4h+1 is a mixed resolving set for WCSn,1. We have the following two cases to be considered:Subcase (i): ifWm⊈k1,k2,k3,…,kn, then there must exist a vertex lj such that lj∈Wm. Then, there exists at least one vertex li∈Wm such that ki−1,ki+1∉Wm. Then, for the corresponding edges vki−1 and vki+1, we have rmvki+1|Wm=rmvki−1|Wm, a contradiction. Therefore, Wm is not a mixed resolving set for WCSn,1 in this case.Subcase (ii): ifWm⊂k1,k2,k3,…,kn, then there exist at least two vertices ki and kj such that ki,kj∉Wm. Then, for the edges vki and vkj, we have rmvki|Wm=rmvkj|Wm, a contradiction. Therefore, Wm is not a mixed resolving set for WCSn,1 in this case as well. Thus, Wm≥4h+1. This completes the proof for the second case.
For rest of the cases, the pattern is the same as that in Case (II).


## 4.1. Cycle Subdivision ofWn,1

SupposeWn,1 is a wheel graph with the vertex set VWn,1=k1,k2,k3,…,kn,v having a single universal vertex v. Now, each cycle edge kjkj+1 of Wn,1 is subdivided with a new vertex lj. The resulting graph so obtained is known as the cycle subdivision wheel graph (CSWG) and is denoted by WCSn,1. CSWG has 3n edges, EWCSn,1=vkj,kjlj,ljkj+1|1≤j≤n, and 2n+1 vertices, VWCSn,1=v,lj,kj|1≤j≤n, where all indices are taken to be modulo n (see Figure 3). In this section, we obtain the mixed metric dimension of CSWG WCSn,1.Figure 3
WCSn,1.Theorem 2.
Forn≥6, we have(6)mdimWCSn,1=4hifn=6h,4h+1ifn=6h+1,4h+2ifn=6h+2,4h+2ifn=6h+3,4h+3ifn=6h+4,4h+4ifn=6h+5.Proof.
To prove this, we first generate the mixed resolving sets for all the cases, obtaining the upper bounds depending on the positive integern. Then, 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≡0mod6. In this case, we have n=6h, where h≥2 and h∈ℕ. Suppose an ordered subset Wm=l1,l2,l4,l5,…,ln−2,ln−1=l3i+1,l3i+2|0≤i≤2h−1 of vertices in WCSn,1 with Wm=4h. Next, we claim that Wm is the mixed resolving set for WCSn,1. Now, we can give mixed codes to every vertex and edge of WCSn,1 with respect to Wm. The sets of mixed metric codes for the vertices u=v,lj,kj|1≤j≤n of WCSn,1 are as follows:(7)A=rmv|Wm=2,2,2,…,2︸4h−times,B=rmkj|Wm=3,3,3,…,3,dl3i+2,k3i+3=1,3,…,3|j≡0mod30≤i≤2h−1∪rmkj|Wm=3,3,3,…,3,dl3i+1,k3i+1=1,3,…,3|j≡1mod30≤i≤2h−1∪rmkj|Wm=3,3,3,…,3,dl3i+1,k3i+2=1,dl3i+2,k3i+2=1,3,…,3|j≡2mod30≤i≤2h−1,C=rmlj|Wm=4,,4,…,4,dl3i+2,l3i+3=2,dl3i+4,l3i+3=2,4,…,4|j≡0mod30≤i≤2h−1∪rmlj|Wm=4,4,4,…,4,dl3i+1,l3i+1=0,dl3i+1,l3i+2=2,4,…,4|j≡1mod30≤i≤2h−1∪rmlj|Wm=4,4,4,…,4,dl3i+1,l3i+1=2,dl3i+1,l3i+2=0,4,…,4|j≡2mod30≤i≤2h−1.Next, the sets of mixed metric codes for the edgesvkj,kjlj,ljkj+1|1≤j≤n of WCSn,1 are as follows:(8)D=rmvkj∣Wm=2,2,2,…,2,dl3i+2,vk3i+3=1,2,…,2∣j≡0mod3&0≤i≤2h−1∪rmvkj∣Wm=2,2,2,…,2,dl3i+1,vk3i+2=1,2,…,2∣j≡1mod3&0≤i≤2h−1∪rmvkj∣Wm=2,2,2,…,2,dl3i+1,vk3i+2=1,dl3i+2,vk3i+2=1,2,…,2∣j≡2mod3&0≤i≤2h−1;E=rmkjlj∣Wm=3,3,3,…,3,dl3i+2,k3i+3l3i+3=1,dl3i+4,k3i+3l3i+3=2,3,…,3∣j≡0mod3&0≤i≤2h−1∪rmkjlj∣Wm=3,3,3,…,3,dl3i+1,k3i+1l3i+1=0,dl3i+2,k3i+1l3i+1=2,3,…,3∣j≡1mod3&0≤i≤2h−1∪rmkjlj∣Wm=3,3,3,…,3,dl3i+1,k3i+2l3i+2=1,dl3i+2,k3i+2l3i+2=0,3,…,3∣j≡2mod3&0≤i≤2h−1;F=rmljkj+1∣Wm=3,3,3,…,3,dl3i+2,l3i+3k3i+4=2,dl3i+4,l3i+3k3i+4=1,3,…,3∣j≡0mod3&0≤i≤2h−1∪rmljkj+1∣Wm=3,3,3,…,3,dl3i+1,l3i+1k3i+2=0,dl3i+2,l3i+1k3i+2=1,3,…,3∣j≡1mod3&0≤i≤2h−1∪rmljkj+1∣Wm=3,3,3,…,3,dl3i+1,l3i+2k3i+3=2,dl3i+2,l3i+2k3i+3=0,3,…,3∣j≡2mod3&0≤i≤2h−1.From these sets of mixed codes forWCSn,1, we obtain that A=1, B=C=D=E=F=n, and A∩B∩C∩D∩E∩F=∅, implying Wm to be a mixed resolving set for WCSn,1, i.e., mdimWCSn,1≤4h. Next, using equation (1) and Proposition 2, we find that mdimWCSn,1=4h, in this case.Case (II):n≡1mod6. In this case, we have n=6h+1, where h≥2 and h∈ℕ. Suppose an ordered subset Wm=l1,l2,l4,l5,…,ln−3,ln−2,ln=l3i+1,l3i+2|0≤i≤2h−1∪ln of vertices in WCSn,1 with Wm=4h+1. Next, we claim that Wm is the mixed resolving set for WCSn,1. Now, we can give mixed codes to every vertex and edge of WCSn,1 with respect to Wm. The sets of mixed metric codes for the vertices u=v,lj,kj|1≤j≤n of WCSn,1 are as follows:(9)A=rmv|Wm=2,2,2,…,2︸4h+1−times;B=rmkj|Wm=3,3,3,…,3,dl3i+2,k3i+3=1,3,…,3|j≡0mod30≤i≤2h−1∪rmk1|Wm=1,3,3,…,3︸4h−1−times,1∪rmkj|Wm=3,3,3,…,3,dl3i+1,k3i+1=1,3,…,3|j≡1mod31≤i≤2h∪rmkj|Wm=3,3,3,…,3,dl3i+1,k3i+2=1,dl3i+2,k3i+2=1,3,…,3|j≡2mod30≤i≤2h−1,C=rmlj|Wm=4,,4,…,4,dl3i+2,l3i+3=2,dl3i+4,l3i+3=2,4,…,4|j≡0mod30≤i≤2h−1∪rml1|Wm=0,2,4,4,4,…,4︸4h−2−times,2∪rmlj|Wm=4,4,4,…,4,dl3i+1,l3i+1=0,dl3i+1,l3i+2=2,4,…,4|j≡1mod31≤i≤2h∪rmlj|Wm=4,4,4,…,4,dl3i+1,l3i+1=2,dl3i+1,l3i+2=0,4,…,4|j≡2mod30≤i≤2h−1.Next, the sets of mixed metric codes for the edgesvkj,kjlj,ljkj+1|1≤j≤n of WCSn,1 are as follows:(10)D=rmvkj|Wm=2,2,2,…,2,dl3i+2,vk3i+3=1,2,…,2|j≡0mod30≤i≤2h−1∪rmvk1|Wm=1,2,2,2,…,2︸4h−1−times,1∪rmvkj|Wm=2,2,2,…,2,dl3i+1,vk3i+1=1,2,…,2|j≡1mod31≤i≤2h∪rmvkj|Wm=2,2,2,…,2,dl3i+1,vk3i+2=1,dl3i+2,vk3i+2=1,2,…,2|j≡2mod30≤i≤2h−1,E=rmkjlj|Wm=3,3,3,…,3,dl3i+2,k3i+3l3i+3=1,dl3i+4,k3i+3l3i+3=2,3,…,3|j≡0mod30≤i≤2h−1∪rmk1l1|Wm=0,2,3,3,3,…,3︸4h−2−times,1∪rmkjlj|Wm=3,3,3,…,3,dl3i+1,k3i+1l3i+1=0,dl3i+2,k3i+1l3i+1=2,3,…,3|j≡1mod31≤i≤2h∪rmkjlj|Wm=3,3,3,…,3,dl3i+1,k3i+2l3i+2=1,dl3i+2,k3i+2l3i+2=0,3,…,3|j≡2mod30≤i≤2h−1,F=rmljkj+1|Wm=3,3,3,…,3,dl3i+2,l3i+3k3i+4=2,dl3i+4,l3i+3k3i+4=1,3,…,3|j≡0mod30≤i≤2h−1∪rml1k2|Wm=0,1,3,3,3,…,3︸4h−2−times,1∪rmljkj+1|Wm=3,3,3,…,3,dl3i+1,l3i+1k3i+2=0,dl3i+2,l3i+1k3i+2=1,3,…,3|j≡1mod31≤i≤2h∪rmljkj+1|Wm=3,3,3,…,3,dl3i+1,l3i+2k3i+3=2,dl3i+2,l3i+2k3i+3=0,3,…,3|j≡2mod30≤i≤2h−1.From these sets of mixed codes forWCSn,1, we obtain that A=1, B=C=D=E=F=n, and A∩B∩C∩D∩E∩F=∅, implying Wm to be a mixed resolving set for WCSn,1, i.e., mdimWCSn,1≤4h+1.Case (III):n≡2mod6. In this case, we have n=6h+2, where h≥2 and h∈ℕ. Suppose an ordered subset Wm=l1,l2,l4,l5,l7…,ln−1,ln=l3i+1,l3i+2|0≤i≤2h of vertices in WCSn,1 with Wm=4h+2. Next, we claim that Wm is the mixed resolving set for WCSn,1. Now, we can give mixed codes to every vertex and edge of WCSn,1 with respect to Wm. The sets of mixed metric codes for the vertices u=v,lj,kj|1≤j≤n of WCSn,1 are as follows:(11)A=rmv|Wm=2,2,2,…,2︸4h+2−times,B=rmkj|Wm=3,3,3,…,3,dl3i+2,k3i+3=1,3,…,3|j≡0mod30≤i≤2h−1∪rmk1|Wm=1,3,3,…,3︸4h−times,1∪rmkj|Wm=3,3,3,…,3,dl3i+1,k3i+1=1,3,…,3|j≡1mod31≤i≤2h∪rmkj|Wm=3,3,3,…,3,dl3i+1,k3i+2=1,dl3i+2,k3i+2=1,3,…,3|j≡2mod30≤i≤2h,C=rmlj|Wm=4,,4,…,4,dl3i+2,l3i+3=2,dl3i+4,l3i+3=2,4,…,4|j≡0mod30≤i≤2h−1∪rml1|Wm=0,2,4,4,4,…,4︸4h−1−times,2∪rmlj|Wm=4,4,4,…,4,dl3i+1,l3i+1=0,dl3i+1,l3i+2=2,4,…,4|j≡1mod31≤i≤2h∪rmlj|Wm=4,4,4,…,4,dl3i+1,l3i+2=2,dl3i+2,l3i+2=0,4,…,4|j≡2mod30≤i≤2h.Next, the sets of mixed metric codes for the edgesvkj,kjlj,ljkj+1|1≤j≤n of WCSn,1 are as follows:(12)D=rmvkj|Wm=2,2,2,…,2,dl3i+2,vk3i+3=1,2,…,2|j≡0mod30≤i≤2h−1∪rmvk1|Wm=1,2,2,2,…,2︸4h−times,1∪rmvkj|Wm=2,2,2,…,2,dl3i+1,vk3i+1=1,2,…,2|j≡1mod31≤i≤2h∪rmvkj|Wm=2,2,2,…,2,dl3i+1,vk3i+2=1,dl3i+2,vk3i+2=1,2,…,2|j≡2mod30≤i≤2h,E=rmkjlj|Wm=3,3,3,…,3,dl3i+2,k3i+3l3i+3=1,dl3i+4,k3i+3l3i+3=2,3,…,3|j≡0mod30≤i≤2h−1∪rmk1l1|Wm=0,2,3,3,3,…,3︸4h−1−times,1∪rmkjlj|Wm=3,3,3,…,3,dl3i+1,k3i+1l3i+1=0,dl3i+2,k3i+1l3i+1=2,3,…,3|j≡1mod31≤i≤2h∪rmkjlj|Wm=3,3,3,…,3,dl3i+1,k3i+2l3i+2=1,dl3i+2,k3i+2l3i+2=0,3,…,3|j≡2mod30≤i≤2h,F=rmljkj+1|Wm=3,3,3,…,3,dl3i+2,l3i+3k3i+4=2,dl3i+4,l3i+3k3i+4=1,3,…,3|j≡0mod30≤i≤2h−1∪rml1k2|Wm=0,1,3,3,3,…,3︸4h−1−times,1∪rmljkj+1|Wm=3,3,3,…,3,dl3i+1,l3i+1k3i+2=0,dl3i+2,l3i+1k3i+2=1,3,…,3|j≡1mod31≤i≤2h∪rmljkj+1|Wm=3,3,3,…,3,dl3i+1,l3i+2k3i+3=2,dl3i+2,l3i+2k3i+3=0,3,…,3|j≡2mod30≤i≤2h.From these sets of mixed codes forWCSn,1, we obtain that A=1, B=C=D=E=F=n, and A∩B∩C∩D∩E∩F=∅, implying Wm to be a mixed resolving set for WCSn,1, i.e., mdimWCSn,1≤4h+2.Case (IV):n≡3mod6. In this case, we have n=6h+3, where h≥2 and h∈ℕ. Suppose an ordered subset Wm=l1,l2,l4,l5,l7…,ln−2,ln−1=l3i+1,l3i+2|0≤i≤2h of vertices in WCSn,1 with Wm=4h+2. Next, we claim that Wm is the mixed resolving set for WCSn,1. Now, we can give mixed codes to every vertex and edge of WCSn,1 with respect to Wm. The sets of mixed metric codes for the vertices u=v,lj,kj|1≤j≤n of WCSn,1 are as follows:(13)A=rmv|Wm=2,2,2,…,2︸4h+2−times,B=rmkj|Wm=3,3,3,…,3,dl3i+2,k3i+3=1,3,…,3|j≡0mod30≤i≤2h∪rmkj|Wm=3,3,3,…,3,dl3i+1,k3i+1=1,3,…,3|j≡1mod30≤i≤2h∪rmkj|Wm=3,3,3,…,3,dl3i+1,k3i+2=1,dl3i+2,k3i+2=1,3,…,3|j≡2mod30≤i≤2h,C=rmlj|Wm=4,,4,…,4,dl3i+2,l3i+3=2,dl3i+4,l3i+3=2,4,…,4|j≡0mod30≤i≤2h∪rmlj|Wm=4,4,4,…,4,dl3i+1,l3i+1=0,dl3i+1,l3i+2=2,4,…,4|j≡1mod30≤i≤2h∪rmlj|Wm=4,4,4,…,4,dl3i+1,l3i+2=2,dl3i+2,l3i+2=0,4,…,4|j≡2mod30≤i≤2h.Next, the sets of mixed metric codes for the edgesvkj,kjlj,ljkj+1|1≤j≤n of WCSn,1 are as follows:(14)D=rmvkj∣Wm=2,2,2,…,2,dl3i+2,vk3i+3=1,2,…,2∣j≡0mod3&0≤i≤2h∪rmvkj∣Wm=2,2,2,…,2,dl3i+1,vk3i+2=1,2,…,2∣j≡1mod3&0≤i≤2h∪rmvkj∣Wm=2,2,2,…,2,dl3i+1,vk3i+2=1,dl3i+2,vk3i+2=1,2,…,2∣j≡2mod3&0≤i≤2h;E=rmkjlj∣Wm=3,3,3,…,3,dl3i+2,k3i+3l3i+3=1,dl3i+4,k3i+3l3i+3=2,3,…,3∣j≡0mod3&0≤i≤2h∪rmkjlj∣Wm=3,3,3,…,3,dl3i+1,k3i+1l3i+1=0,dl3i+2,k3i+1l3i+1=2,3,…,3∣j≡1mod3&0≤i≤2h∪rmkjlj∣Wm=3,3,3,…,3,dl3i+1,k3i+2l3i+2=1,dl3i+2,k3i+2l3i+2=0,3,…,3∣j≡2mod3&0≤i≤2h;F=rmljkj+1∣Wm=3,3,3,…,3,dl3i+2,l3i+3k3i+4=2,dl3i+4,l3i+3k3i+4=1,3,…,3∣j≡0mod3&0≤i≤2h∪rmljkj+1∣Wm=3,3,3,…,3,dl3i+1,l3i+1k3i+2=0,dl3i+2,l3i+1k3i+2=1,3,…,3∣j≡1mod3&0≤i≤2h.∪rmljkj+1∣Wm=3,3,3,…,3,dl3i+1,l3i+2k3i+3=2,dl3i+2,l3i+2k3i+3=0,3,…,3∣j≡2mod3&0≤i≤2h;From these sets of mixed codes forWCSn,1, we obtain that A=1, B=C=D=E=F=n, and A∩B∩C∩D∩E∩F=∅, implying Wm to be a mixed resolving set for WCSn,1, i.e., mdimWCSn,1≤4h+2. Next, using equation (1) and Proposition 2, we find that mdimWCSn,1=4h+2, in this case.Case (V):n≡4mod6. In this case, we have n=6h+4, where h≥2 and h∈ℕ. Suppose an ordered subset Wm=l1,l2,l4,l5,…,ln−3,ln−2,ln=l3i+1,l3i+2|0≤i≤2h∪ln of vertices in WCSn,1 with Wm=4h+3. Next, we claim that Wm is the mixed resolving set for WCSn,1. Now, we can give mixed codes to every vertex and edge of WCSn,1 with respect to Wm. The sets of mixed metric codes for the vertices u=v,lj,kj|1≤j≤n of WCSn,1 are as follows:(15)A=rmv|Wm=2,2,2,…,2︸4h+3−times,B=rmkj|Wm=3,3,3,…,3,dl3i+2,k3i+3=1,3,…,3|j≡0mod30≤i≤2h∪rmk1|Wm=1,3,3,…,3︸4h+1−times,1∪rmkj|Wm=3,3,3,…,3,dl3i+1,k3i+1=1,3,…,3|j≡1mod31≤i≤2h+1∪rmkj|Wm=3,3,3,…,3,dl3i+1,k3i+2=1,dl3i+2,k3i+2=1,3,…,3|j≡2mod30≤i≤2h+1,C=rmlj|Wm=4,,4,…,4,dl3i+2,l3i+3=2,dl3i+4,l3i+3=2,4,…,4|j≡0mod30≤i≤2h∪rml1|Wm=0,2,4,4,4,…,4︸4h−times,2∪rmlj|Wm=4,4,4,…,4,dl3i+1,l3i+1=0,dl3i+1,l3i+2=2,4,…,4|j≡1mod31≤i≤2h+1∪rmlj|Wm=4,4,4,…,4,dl3i+1,l3i+1=2,dl3i+1,l3i+2=0,4,…,4|j≡2mod30≤i≤2h+1.Next, the sets of mixed metric codes for the edgesvkj,kjlj,ljkj+1|1≤j≤n of WCSn,1 are as follows:(16)D=rmvkj|Wm=2,2,2,…,2,dl3i+2,vk3i+3=1,2,…,2|j≡0mod30≤i≤2h∪rmvk1|Wm=1,2,2,2,…,2︸4h+1−times,1∪rmvkj|Wm=2,2,2,…,2,dl3i+1,vk3i+1=1,2,…,2|j≡1mod31≤i≤2h+1∪rmvkj|Wm=2,2,2,…,2,dl3i+1,vk3i+2=1,dl3i+2,vk3i+2=1,2,…,2|j≡2mod30≤i≤2h,E=rmkjlj|Wm=3,3,3,…,3,dl3i+2,k3i+3l3i+3=1,dl3i+4,k3i+3l3i+3=2,3,…,3|j≡0mod30≤i≤2h∪rmk1l1|Wm=0,2,3,3,3,…,3︸4h−times,1∪rmkjlj|Wm=3,3,3,…,3,dl3i+1,k3i+1l3i+1=0,dl3i+2,k3i+1l3i+1=2,3,…,3|j≡1mod31≤i≤2h+1∪rmkjlj|Wm=3,3,3,…,3,dl3i+1,k3i+2l3i+2=1,dl3i+2,k3i+2l3i+2=0,3,…,3|j≡2mod30≤i≤2h+1,F=rmljkj+1|Wm=3,3,3,…,3,dl3i+2,l3i+3k3i+4=2,dl3i+4,l3i+3k3i+4=1,3,…,3|j≡0mod30≤i≤2h∪rml1k2|Wm=0,1,3,3,3,…,3︸4h−times,1∪rmljkj+1|Wm=3,3,3,…,3,dl3i+1,l3i+1k3i+2=0,dl3i+2,l3i+1k3i+2=1,3,…,3|j≡1mod31≤i≤2h+1∪rmljkj+1|Wm=3,3,3,…,3,dl3i+1,l3i+2k3i+3=2,dl3i+2,l3i+2k3i+3=0,3,…,3|j≡2mod30≤i≤2h.From these sets of mixed codes forWCSn,1, we obtain that A=1,B=C=D=E=F=n, and A∩B∩C∩D∩E∩F=∅, implying Wm to be a mixed resolving set for WCSn,1, i.e., mdimWCSn,1≤4h+3.Case (VI):n≡5mod6. In this case, we have n=6h+5, where h≥1 and h∈ℕ. Suppose an ordered subset Wm=l1,l2,l4,l5,…,ln−1,ln=l3i+1,l3i+2|0≤i≤2h+1 of vertices in WCSn,1 with Wm=4h+4. Next, we claim that Wm is the mixed resolving set for WCSn,1. Now, we can give mixed codes to every vertex and edge of WCSn,1 with respect to Wm. The sets of mixed metric codes for the vertices u=v,lj,kj|1≤j≤n of WCSn,1 are as follows:(17)A=rmv|Wm=2,2,2,…,2︸4h+4−times,B=rmkj|Wm=3,3,3,…,3,dl3i+2,k3i+3=1,3,…,3|j≡0mod30≤i≤2h∪rmk1|Wm=1,3,3,…,3︸4h+2−times,1∪rmkj|Wm=3,3,3,…,3,dl3i+1,k3i+1=1,3,…,3|j≡1mod31≤i≤2h+1∪rmkj|Wm=3,3,3,…,3,dl3i+1,k3i+2=1,dl3i+2,k3i+2=1,3,…,3|j≡2mod30≤i≤2h+1,C=rmlj|Wm=4,,4,…,4,dl3i+2,l3i+3=2,dl3i+4,l3i+3=2,4,…,4|j≡0mod30≤i≤2h∪rml1|Wm=0,2,4,4,4,…,4︸4h+1−times,2∪rmlj|Wm=4,4,4,…,4,dl3i+1,l3i+1=0,dl3i+1,l3i+2=2,4,…,4|j≡1mod31≤i≤2h+1∪rmlj|Wm=4,4,4,…,4,dl3i+1,l3i+2=2,dl3i+2,l3i+2=0,4,…,4|j≡2mod30≤i≤2h+1.Next, the sets of mixed metric codes for the edgesvkj,kjlj,ljkj+1|1≤j≤n of WCSn,1 are as follows:(18)D=rmvkj|Wm=2,2,2,…,2,dl3i+2,vk3i+3=1,2,…,2|j≡0mod30≤i≤2h∪rmvk1|Wm=1,2,2,2,…,2︸4h+2−times,1∪rmvkj|Wm=2,2,2,…,2,dl3i+1,vk3i+1=1,2,…,2|j≡1mod31≤i≤2h+1∪rmvkj|Wm=2,2,2,…,2,dl3i+1,vk3i+2=1,dl3i+2,vk3i+2=1,2,…,2|j≡2mod30≤i≤2h+1,E=rmkjlj|Wm=3,3,3,…,3,dl3i+2,k3i+3l3i+3=1,dl3i+4,k3i+3l3i+3=2,3,…,3|j≡0mod30≤i≤2h∪rmk1l1|Wm=0,2,3,3,3,…,3︸4h+1−times,1∪rmkjlj|Wm=3,3,3,…,3,dl3i+1,k3i+1l3i+1=0,dl3i+2,k3i+1l3i+1=2,3,…,3|j≡1mod31≤i≤2h+1∪rmkjlj|Wm=3,3,3,…,3,dl3i+1,k3i+2l3i+2=1,dl3i+2,k3i+2l3i+2=0,3,…,3|j≡2mod30≤i≤2h+1,F=rmljkj+1|Wm=3,3,3,…,3,dl3i+2,l3i+3k3i+4=2,dl3i+4,l3i+3k3i+4=1,3,…,3|j≡0mod30≤i≤2h∪rml1k2|Wm=0,1,3,3,3,…,3︸4h+1−times,2∪rmljkj+1|Wm=3,3,3,…,3,dl3i+1,l3i+1k3i+2=0,dl3i+2,l3i+1k3i+2=1,3,…,3|j≡1mod31≤i≤2h+1∪rmljkj+1|Wm=3,3,3,…,3,dl3i+1,l3i+2k3i+3=2,dl3i+2,l3i+2k3i+3=0,3,…,3|j≡2mod30≤i≤2h+1.From these sets of mixed codes forWCSn,1, we obtain that A=1, B=C=D=E=F=n, and A∩B∩C∩D∩E∩F=∅, implying Wm to be a mixed resolving set for WCSn,1, i.e., mdimWCSn,1≤4h+4. Now, for the second, third, fifth, and sixth case, we obtain their lower bounds as follows.For the second case, suppose thatWm⊂VWCSn,1 with Wm<4h+1 is a mixed resolving set for WCSn,1. We have the following two cases to be considered:Subcase (i): ifWm⊈k1,k2,k3,…,kn, then there must exist a vertex lj such that lj∈Wm. Then, there exists at least one vertex li∈Wm such that ki−1,ki+1∉Wm. Then, for the corresponding edges vki−1 and vki+1, we have rmvki+1|Wm=rmvki−1|Wm, a contradiction. Therefore, Wm is not a mixed resolving set for WCSn,1 in this case.Subcase (ii): ifWm⊂k1,k2,k3,…,kn, then there exist at least two vertices ki and kj such that ki,kj∉Wm. Then, for the edges vki and vkj, we have rmvki|Wm=rmvkj|Wm, a contradiction. Therefore, Wm is not a mixed resolving set for WCSn,1 in this case as well. Thus, Wm≥4h+1. This completes the proof for the second case.
For rest of the cases, the pattern is the same as that in Case (II).


## 5. Mixed Metric Dimension of the Barycentric Subdivision ofWn,1

In this section, we determine the mixed metric dimension of the barycentric subdivision of a wheel graph.


### 5.1. Barycentric Subdivision ofWn,1

SupposeWn,1 is a wheel graph with the vertex set VWn,1=k1,k2,k3,…,kn,v having a single universal vertex v. Now, each of the edges kjkj+1 and vkj1≤j≤n of Wn,1 is subdivided with a new vertex. The resulting graph so obtained is known as the barycentric subdivision wheel graph (BSWG) and is denoted by WBSn,1. BSWG has 4n edges, EWBSn,1=vlj,ljkj,kjmj,mjkj+1|1≤j≤n, and 3n+1 vertices, VWBSn,1=v,lj,kj,mj|1≤j≤n, where all indices are taken to be modulo n (see Figure 4). In this section, we obtain the mixed metric dimension of BSWG WBSn,1.Figure 4
WBSn,1.Theorem 3.
Forn≥6, we have(19)mdimWBSn,1=4hifn=6h,4h+1ifn=6h+1,4h+2ifn=6h+2,4h+2ifn=6h+3,4h+3ifn=6h+4,4h+4ifn=6h+5.Proof.
To prove this, we first generate the mixed resolving sets for all the cases, obtaining the upper bounds depending on the positive integern. Then, 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≡0mod6. In this case, we have n=6h, where h≥2 and h∈ℕ. Suppose an ordered subset Wm=m1,m2,m4,m5,…,mn−2,mn−1=m3i+1,m3i+2|0≤i≤2h−1 of vertices in WBSn,1 with Wm=4h. Next, we claim that Wm is the mixed resolving set for WBSn,1. Now, we can give mixed codes to every vertex and edge of WBSn,1 with respect to Wm. The sets of mixed metric codes for the vertices u=v,kj,lj,mj|1≤j≤n of WBSn,1 are as follows:(20)A=rmv|Wm=3,3,3,…,3︸4h−times,B=rmkj|Wm=5,5,5,…,5,dm3i+1,k3i+3=3,dm3i+2,k3i+3=1,dm3i+4,k3i+3=5,5,…,5|j≡0mod30≤i≤2h−1∪rmk1|Wm=1,3,5,…,5︸4h−3−times,3∪rmkj|Wm=5,5,5,…,5,dm3i+2,k3i+1=3,dm3i+4,k3i+1=1,dm3i+5,k3i+1=3,5,…,5|j≡1mod31≤i≤2h−1∪rmkj|Wm=5,5,5,…,5,dm3i+1,k3i+2=1,dm3i+2,k3i+2=1,5,…,5|j≡2mod30≤i≤2h−1,C=rmlj|Wm=4,,4,…,4,dm3i+2,l3i+3=2,4,…,4|j≡0mod30≤i≤2h−1∪rmlj|Wm=4,4,4,…,4,dm3i+1,l3i+1=2,4,…,4|j≡1mod30≤i≤2h−1∪rmlj|Wm=4,4,4,…,4,dm3i+1,l3i+2=2,dm3i+2,l3i+2=2,4,…,4|j≡2mod30≤i≤2h−1,D=rmmj|Wm=6,,6,…,6,dm3i+1,m3i+3=4,dm3i+2,m3i+3=2,dm3i+4,m3i+3=2,dm3i+5,m3i+3=4,6,…,6|j≡0mod30≤i≤2h−1∪rmm1|Wm=0,2,6,6,…,6︸4h−3−times,4∪rmmj|Wm=6,6,6,…,6,dm3i+2,m3i+1=4,dm3i+4,m3i+1=0,dm3i+5,m3i+1=2,6,…,6|j≡1mod30≤i≤2h−1∪rmlj|Wm=6,6,6,…,6,dm3i+1,m3i+2=2,dm3i+2,m3i+2=0,dm3i+4,m3i+2=4,6,…,6|j≡2mod30≤i≤2h−1.Next, the sets of mixed metric codes for the edgesvlj,ljkj,kjmj,mjkj+1|1≤j≤n of WBSn,1 are as follows:(21)E=rmvlj|Wm=3,3,3,…,3,dm3i+2,vl3i+3=2,3,…,3|j≡0mod30≤i≤2h−1∪rmvlj|Wm=3,3,3,…,3,dm3i+1,vl3i+1=2,3,…,3|j≡1mod30≤i≤2h−1∪rmvlj|Wm=3,3,3,…,3,dm3i+1,vl3i+2=2,dm3i+2,vl3i+2=2,3,…,3|j≡2mod30≤i≤2h−1,F=rmljkj|Wm=4,4,4,…,4,dm3i+1,l3i+3k3i+3=3,dm3i+2,l3i+3k3i+3=1,dm3i+4,l3i+3k3i+3=3,4,…,4|j≡0mod30≤i≤2h−1∪rml1k1|Wm=1,3,4,4,…,4︸4h−3−times,3∪rmljkj|Wm=4,4,4,…,4,dm3i+2,l3i+1k3i+1=3,dm3i+4,l3i+1k3i+1=1,dm3i+5,l3i+1k3i+1=3,4,…,4|j≡1mod31≤i≤2h−1∪rmljkj|Wm=4,4,4,…,4,dm3i+1,l3i+2k3i+2=1,dm3i+2,l3i+2l3i+2=1,4,…,4|j≡2mod30≤i≤2h−1,G=rmkjmj|Wm=5,5,5,…,5,dm3i+1,k3i+3m3i+3=3,dm3i+2,k3i+3m3i+3=1,dm3i+4,k3i+3m3i+3=2,dm3i+5,k3i+3m3i+3=4,5,…,5|j≡0mod30≤i≤2h−1∪rmk1m1|Wm=0,2,5,5,…,5︸4h−3−times,3∪rmkjmj|Wm=5,5,5,…,5,dm3i+2,k3i+1m3i+1=3,dm3i+4,k3i+1m3i+1=0,dm3i+5,k3i+1m3i+1=2,5,…,5|j≡1mod31≤i≤2h−1∪rmkjmj|Wm=5,5,5,…,5,dm3i+1,k3i+2m3i+2=1,dm3i+2,k3i+2m3i+2=0,dm3i+4,k3i+2m3i+2=4,5,…,5|j≡2mod30≤i≤2h−1,H=rmmjkj+1|Wm=5,5,5,…,5,dm3i+1,m3i+3k3i+4=4,dm3i+2,m3i+3k3i+4=2,dm3i+4,m3i+3k3i+4=1,dm3i+5,m3i+3k3i+4=3,5,…,5|j≡0mod30≤i≤2h−1∪rmm1k2|Wm=0,1,5,5,…,5︸4h−3−times,4∪rmmjkj+1|Wm=5,5,5,…,5,dm3i+2,m3i+1k3i+2=4,dm3i+4,m3i+1k3i+2=0,dm3i+5,m3i+1k3i+2=1,5,…,5|j≡1mod31≤i≤2h−1∪rmmjkj+1|Wm=5,5,5,…,5,dm3i+1,m3i+2k3i+3=2,dm3i+2,m3i+2k3i+3=0,dm3i+4,m3i+2k3i+3=3,5,…,5|j≡2mod30≤i≤2h−1.From these sets of mixed codes forWBSn,1, we obtain that A=1, B=C=D=E=F=G=H=n, and A∩B∩C∩D∩E∩F∩G∩H=∅, implying Wm to be a mixed resolving set for WBSn,1, i.e., mdimWBSn,1≤4h. Next, using equation (1) and Proposition 2, we find that mdimWBSn,1=4h, in this case.
Like the first case, the rest of the proof is similar to that of Theorem2.Remark 2.
For the cycle and barycentric subdivision wheel graph, i.e.,H=WCSn,1 and H=WBSn,1, we find that dimH=edimH=mdimH when n=6h and n=6h+3. For the rest of the values of the positive integer n, we have dimH=edimH<mdimH (using Propositions 2 and 4 and Theorems 2 and 3).


## 5.1. Barycentric Subdivision ofWn,1

SupposeWn,1 is a wheel graph with the vertex set VWn,1=k1,k2,k3,…,kn,v having a single universal vertex v. Now, each of the edges kjkj+1 and vkj1≤j≤n of Wn,1 is subdivided with a new vertex. The resulting graph so obtained is known as the barycentric subdivision wheel graph (BSWG) and is denoted by WBSn,1. BSWG has 4n edges, EWBSn,1=vlj,ljkj,kjmj,mjkj+1|1≤j≤n, and 3n+1 vertices, VWBSn,1=v,lj,kj,mj|1≤j≤n, where all indices are taken to be modulo n (see Figure 4). In this section, we obtain the mixed metric dimension of BSWG WBSn,1.Figure 4
WBSn,1.Theorem 3.
Forn≥6, we have(19)mdimWBSn,1=4hifn=6h,4h+1ifn=6h+1,4h+2ifn=6h+2,4h+2ifn=6h+3,4h+3ifn=6h+4,4h+4ifn=6h+5.Proof.
To prove this, we first generate the mixed resolving sets for all the cases, obtaining the upper bounds depending on the positive integern. Then, 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≡0mod6. In this case, we have n=6h, where h≥2 and h∈ℕ. Suppose an ordered subset Wm=m1,m2,m4,m5,…,mn−2,mn−1=m3i+1,m3i+2|0≤i≤2h−1 of vertices in WBSn,1 with Wm=4h. Next, we claim that Wm is the mixed resolving set for WBSn,1. Now, we can give mixed codes to every vertex and edge of WBSn,1 with respect to Wm. The sets of mixed metric codes for the vertices u=v,kj,lj,mj|1≤j≤n of WBSn,1 are as follows:(20)A=rmv|Wm=3,3,3,…,3︸4h−times,B=rmkj|Wm=5,5,5,…,5,dm3i+1,k3i+3=3,dm3i+2,k3i+3=1,dm3i+4,k3i+3=5,5,…,5|j≡0mod30≤i≤2h−1∪rmk1|Wm=1,3,5,…,5︸4h−3−times,3∪rmkj|Wm=5,5,5,…,5,dm3i+2,k3i+1=3,dm3i+4,k3i+1=1,dm3i+5,k3i+1=3,5,…,5|j≡1mod31≤i≤2h−1∪rmkj|Wm=5,5,5,…,5,dm3i+1,k3i+2=1,dm3i+2,k3i+2=1,5,…,5|j≡2mod30≤i≤2h−1,C=rmlj|Wm=4,,4,…,4,dm3i+2,l3i+3=2,4,…,4|j≡0mod30≤i≤2h−1∪rmlj|Wm=4,4,4,…,4,dm3i+1,l3i+1=2,4,…,4|j≡1mod30≤i≤2h−1∪rmlj|Wm=4,4,4,…,4,dm3i+1,l3i+2=2,dm3i+2,l3i+2=2,4,…,4|j≡2mod30≤i≤2h−1,D=rmmj|Wm=6,,6,…,6,dm3i+1,m3i+3=4,dm3i+2,m3i+3=2,dm3i+4,m3i+3=2,dm3i+5,m3i+3=4,6,…,6|j≡0mod30≤i≤2h−1∪rmm1|Wm=0,2,6,6,…,6︸4h−3−times,4∪rmmj|Wm=6,6,6,…,6,dm3i+2,m3i+1=4,dm3i+4,m3i+1=0,dm3i+5,m3i+1=2,6,…,6|j≡1mod30≤i≤2h−1∪rmlj|Wm=6,6,6,…,6,dm3i+1,m3i+2=2,dm3i+2,m3i+2=0,dm3i+4,m3i+2=4,6,…,6|j≡2mod30≤i≤2h−1.Next, the sets of mixed metric codes for the edgesvlj,ljkj,kjmj,mjkj+1|1≤j≤n of WBSn,1 are as follows:(21)E=rmvlj|Wm=3,3,3,…,3,dm3i+2,vl3i+3=2,3,…,3|j≡0mod30≤i≤2h−1∪rmvlj|Wm=3,3,3,…,3,dm3i+1,vl3i+1=2,3,…,3|j≡1mod30≤i≤2h−1∪rmvlj|Wm=3,3,3,…,3,dm3i+1,vl3i+2=2,dm3i+2,vl3i+2=2,3,…,3|j≡2mod30≤i≤2h−1,F=rmljkj|Wm=4,4,4,…,4,dm3i+1,l3i+3k3i+3=3,dm3i+2,l3i+3k3i+3=1,dm3i+4,l3i+3k3i+3=3,4,…,4|j≡0mod30≤i≤2h−1∪rml1k1|Wm=1,3,4,4,…,4︸4h−3−times,3∪rmljkj|Wm=4,4,4,…,4,dm3i+2,l3i+1k3i+1=3,dm3i+4,l3i+1k3i+1=1,dm3i+5,l3i+1k3i+1=3,4,…,4|j≡1mod31≤i≤2h−1∪rmljkj|Wm=4,4,4,…,4,dm3i+1,l3i+2k3i+2=1,dm3i+2,l3i+2l3i+2=1,4,…,4|j≡2mod30≤i≤2h−1,G=rmkjmj|Wm=5,5,5,…,5,dm3i+1,k3i+3m3i+3=3,dm3i+2,k3i+3m3i+3=1,dm3i+4,k3i+3m3i+3=2,dm3i+5,k3i+3m3i+3=4,5,…,5|j≡0mod30≤i≤2h−1∪rmk1m1|Wm=0,2,5,5,…,5︸4h−3−times,3∪rmkjmj|Wm=5,5,5,…,5,dm3i+2,k3i+1m3i+1=3,dm3i+4,k3i+1m3i+1=0,dm3i+5,k3i+1m3i+1=2,5,…,5|j≡1mod31≤i≤2h−1∪rmkjmj|Wm=5,5,5,…,5,dm3i+1,k3i+2m3i+2=1,dm3i+2,k3i+2m3i+2=0,dm3i+4,k3i+2m3i+2=4,5,…,5|j≡2mod30≤i≤2h−1,H=rmmjkj+1|Wm=5,5,5,…,5,dm3i+1,m3i+3k3i+4=4,dm3i+2,m3i+3k3i+4=2,dm3i+4,m3i+3k3i+4=1,dm3i+5,m3i+3k3i+4=3,5,…,5|j≡0mod30≤i≤2h−1∪rmm1k2|Wm=0,1,5,5,…,5︸4h−3−times,4∪rmmjkj+1|Wm=5,5,5,…,5,dm3i+2,m3i+1k3i+2=4,dm3i+4,m3i+1k3i+2=0,dm3i+5,m3i+1k3i+2=1,5,…,5|j≡1mod31≤i≤2h−1∪rmmjkj+1|Wm=5,5,5,…,5,dm3i+1,m3i+2k3i+3=2,dm3i+2,m3i+2k3i+3=0,dm3i+4,m3i+2k3i+3=3,5,…,5|j≡2mod30≤i≤2h−1.From these sets of mixed codes forWBSn,1, we obtain that A=1, B=C=D=E=F=G=H=n, and A∩B∩C∩D∩E∩F∩G∩H=∅, implying Wm to be a mixed resolving set for WBSn,1, i.e., mdimWBSn,1≤4h. Next, using equation (1) and Proposition 2, we find that mdimWBSn,1=4h, in this case.
Like the first case, the rest of the proof is similar to that of Theorem2.Remark 2.
For the cycle and barycentric subdivision wheel graph, i.e.,H=WCSn,1 and H=WBSn,1, we find that dimH=edimH=mdimH when n=6h and n=6h+3. For the rest of the values of the positive integer n, we have dimH=edimH<mdimH (using Propositions 2 and 4 and Theorems 2 and 3).


## 6. Conclusion

In this article, we have computed the mixed metric dimension for three families of graphs, namely,WBSn,1, WCSn,1, and WSSn,1, obtained after the barycentric, cycle, and spoke subdivisions of the wheel graph Wn,1, respectively. We also observed that the mixed resolving sets for WBSn,1 and WCSn,1 are independent. For WSSn,1, we found that dimWSSn,1<edimWSSn,1<mdimWSSn,1, and for H=WBSn,1 and H=WCSn,1, we obtained the following relation: dimH=edimH≤mdimH (partial answers to the questions raised in [1, 18]).



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