Abstract

A vertex distinguishes (or resolves) two elements (edges or vertices) if . A set of vertices in a nontrivial connected graph is said to be a mixed resolving set for if every two different elements (edges and vertices) of are distinguished by at least one vertex of . The mixed resolving set with minimum cardinality in is called the mixed metric dimension (vertex-edge resolvability) of and denoted by . 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.

1. Introduction

Suppose is a nontrivial, simple, and connected graph, where represents a set of edges and represents a set of vertices. The distance between two vertices and in an undirected graph , denoted by , is the length of a shortest path in . In [1], Kelenc et al. introduced the concept of mixed metric dimension in graphs. This dimension of graph is the mixture of metric and edge metric dimensions.

A vertex is said to resolve two vertices and in if . Let be a vertex and be an ordered subset of vertices in . The metric coordinate (or metric representation) of with respect to is the -tuple . Then, is said to be a resolving set (or metric generator) for if for every pair of vertices with , we have . A resolving set with minimum cardinality is called the of , and the cardinality of the metric basis set is the metric dimension of .

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 edge and a vertex is defined as . A vertex is said to resolve two edges and in if . Let be an edge and be an ordered subset of vertices in . The edge metric codes of with respect to are the -tuple . Then, is said to be an edge resolving set for if for every pair of edges with , we have . An edge resolving set with minimum cardinality is called an edge metric basis for , and the cardinality of this edge metric basis set is the edge metric dimension of .

For a connected graph , we see that every vertex of is uniquely recognized by a resolving set of , and every edge of is uniquely recognized by an edge resolving set of ; the natural question is as follows: whether every resolving set is also an edge resolving set for and vice versa? Kelenc et al. in [18] proved that there exist some families of graphs for which the resolving set is also an edge resolving set , but in general, this is not true for every graph . Similarly, for every graph , the edge resolving set is not necessarily a resolving set for .

Let us define a set of elements as , i.e., each element is an edge or a vertex. A vertex is said to resolve two elements and from if . Let be an element and be an ordered subset of vertices in . The mixed metric codes of with respect to are the -tuple . Then, is said to be a mixed resolving set for if for every pair of distinct elements , we have . A mixed resolving set with minimum cardinality is called a mixed metric basis for , and the cardinality of this mixed metric basis set is the mixed metric dimension of . 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

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 . 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 .

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 vertex in an undirected graph is said to be the universal vertex if it is adjacent to all other vertices of . A wheel graph () is a graph with vertices obtained by joining a single universal vertex to all of the vertices of a cycle graph . has a vertex set and an edge set , where all of the indices are taken to be modulo . The edges are called the cycle edges of , and the edges are called as the spokes of the wheel graph.

We state that a family of nontrivial connected graphs has bounded mixed metric dimension if there exists a constant for every graph of such that ; otherwise, has an unbounded mixed metric dimension. If all of the graphs in have the same mixed metric dimension, then is referred to as a family with a constant mixed metric dimension. Cycles and paths for are the graph families with a constant mixed metric dimension.

2.2. Independent Mixed Resolving Set

A set of vertices from is said to be an independent mixed resolving set for if is an independent as well as mixed resolving set.

Let , , and be the graphs obtained from the wheel graph after spoke, cycle, and barycentric subdivisions of , 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]). , for .

Proposition 2 (see [21]). For , we have

Proposition 3 (see [22]). , for .

Proposition 4 (see [23, 22]). For , we have

This article is organized as follows: in Section 3, we will study the mixed metric dimension of the spoke subdivision of the wheel graph . In Sections 4 and 5, we will study the mixed metric dimension of the cycle and barycentric subdivision of the wheel graph, i.e., and , 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 of

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

3.1. Spoke Subdivision of

Suppose is a wheel graph with the vertex set having a single universal vertex . Now, each central spoke of is subdivided with a new vertex . The resulting graph so obtained is known as the spoke subdivision wheel graph (SSWG) and is denoted by . SSWG has edges, , and vertices, , where all indices are taken to be modulo (see Figure 1). In this section, we obtain the mixed metric dimension of SSWG .

Figure 1 

.

Theorem 1. , for .

Proof. To prove that , we construct a mixed resolving set for . Suppose having cycle vertices from . We claim that is a mixed resolving set for . Now, we can give mixed codes to each of the vertex and edge of with respect to .
The sets of mixed metric codes for the vertices of are as follows:Next, the sets of mixed metric codes for the edges of are as follows:From these sets of mixed codes for , we obtain that , , and , implying to be a mixed resolving set for , i.e., . Conversely, suppose, on the contrary, that there exists a mixed resolving set such that . Then, we have the following cases to be considered: Case (i): . In this case, we further have two subcases: Subcase (i): if , then there exists at least one vertex such that . Then, for an edge and the vertex , we have , a contradiction. Therefore, the set is not a mixed resolving set for . Subcase (ii): if , then at least one vertex belongs to the set . Then, there exists one , and the corresponding vertex . Then, for an edge and the vertex , we have , a contradiction. Therefore, again, in this case, the set is not a mixed resolving set for . Case (ii): . In this case, we have two subcases: Subcase (i): if , then there exists at least one vertex such that . Then, clearly, for an edge and the vertex , we have , a contradiction. Therefore, the set is not a mixed resolving set for . Subcase (ii): if at least one must belong to the set , then there exists at least one vertex , and the corresponding vertex . Then, for an edge and a vertex , we have , a contradiction. Therefore, again, in this case, the set is not a mixed resolving set for . Thus, in all the cases, we have , implying , which completes the proof of the theorem.

Remark 1. For the spoke subdivision wheel graph , we find that (using Propositions 1 and 3 and Theorem 1). The comparison between these three dimensions of is clearly shown in Figure 2, and the value of each dimension depends on the number of vertices in .

Figure 2 

Comparison between , , and .

4. Mixed Metric Dimension of the Cycle Subdivision of

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

4.1. Cycle Subdivision of

Suppose is a wheel graph with the vertex set having a single universal vertex . Now, each cycle edge of is subdivided with a new vertex . The resulting graph so obtained is known as the cycle subdivision wheel graph (CSWG) and is denoted by . CSWG has edges, , and vertices, , where all indices are taken to be modulo (see Figure 3). In this section, we obtain the mixed metric dimension of CSWG .

Figure 3 

.

Theorem 2. For , we have

Proof. To prove this, we first generate the mixed resolving sets for all the cases, obtaining the upper bounds depending on the positive integer . 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): . In this case, we have , where and . Suppose an ordered subset of vertices in with . Next, we claim that is the mixed resolving set for . Now, we can give mixed codes to every vertex and edge of with respect to . The sets of mixed metric codes for the vertices of are as follows: Next, the sets of mixed metric codes for the edges of are as follows: From these sets of mixed codes for , we obtain that , , and , implying to be a mixed resolving set for , i.e., . Next, using equation (1) and Proposition 2, we find that , in this case. Case (II): . In this case, we have , where and . Suppose an ordered subset of vertices in with . Next, we claim that is the mixed resolving set for . Now, we can give mixed codes to every vertex and edge of with respect to . The sets of mixed metric codes for the vertices of are as follows: Next, the sets of mixed metric codes for the edges of are as follows: From these sets of mixed codes for , we obtain that , , and , implying to be a mixed resolving set for , i.e., . Case (III): . In this case, we have , where and . Suppose an ordered subset of vertices in with . Next, we claim that is the mixed resolving set for . Now, we can give mixed codes to every vertex and edge of with respect to . The sets of mixed metric codes for the vertices of are as follows: Next, the sets of mixed metric codes for the edges of are as follows: From these sets of mixed codes for , we obtain that , , and , implying to be a mixed resolving set for , i.e., . Case (IV): . In this case, we have , where and . Suppose an ordered subset of vertices in with . Next, we claim that is the mixed resolving set for . Now, we can give mixed codes to every vertex and edge of with respect to . The sets of mixed metric codes for the vertices of are as follows: Next, the sets of mixed metric codes for the edges of are as follows: From these sets of mixed codes for , we obtain that , , and , implying to be a mixed resolving set for , i.e., . Next, using equation (1) and Proposition 2, we find that , in this case. Case (V): . In this case, we have , where and . Suppose an ordered subset of vertices in with . Next, we claim that is the mixed resolving set for . Now, we can give mixed codes to every vertex and edge of with respect to . The sets of mixed metric codes for the vertices of are as follows: Next, the sets of mixed metric codes for the edges of are as follows: From these sets of mixed codes for , we obtain that , and , implying to be a mixed resolving set for , i.e., . Case (VI): . In this case, we have , where and . Suppose an ordered subset of vertices in with . Next, we claim that is the mixed resolving set for . Now, we can give mixed codes to every vertex and edge of with respect to . The sets of mixed metric codes for the vertices of are as follows: Next, the sets of mixed metric codes for the edges of are as follows: From these sets of mixed codes for , we obtain that , , and , implying to be a mixed resolving set for , i.e., . Now, for the second, third, fifth, and sixth case, we obtain their lower bounds as follows. For the second case, suppose that with is a mixed resolving set for . We have the following two cases to be considered: Subcase (i): if , then there must exist a vertex such that . Then, there exists at least one vertex such that . Then, for the corresponding edges and , we have , a contradiction. Therefore, is not a mixed resolving set for in this case. Subcase (ii): if , then there exist at least two vertices and such that . Then, for the edges and , we have , a contradiction. Therefore, is not a mixed resolving set for in this case as well. Thus, . This completes the proof for the second case.For rest of the cases, the pattern is the same as that in Case (II).