Metric, edge-metric, mixed-metric, and fault-tolerant metric dimensions of geometric networks with potential applications

Abstract

Resolvability parameters of graphs are widely applicable in fields like computer science, chemistry, and geography. Many of these parameters, such as the metric dimension, are computationally hard to determine. This paper focuses on Möbius-type geometric graphs, including hexagonal Möbius graphs, barycentric subdivisions of Möbius ladders, and octagonal Möbius chains. It addresses several open problems related to their resolvability parameters, such as the exchange property and fault-tolerant metric dimension. Additionally, the study extends to 30 lower benzenoid hydrocarbons, computing their metric, edge metric, and mixed metric dimensions. These results support structure-property modeling, particularly for predicting the total -electronic energy of such hydrocarbons. The paper concludes with new questions prompted by the findings.

Introduction

A graph consists of nodes/vertices and edges, and an important concept in graph theory is distance. Distance can be found either between nodes or edges of a graph. It is a useful tool for studying various graph properties. An important parameter related to distance is the metric dimension of a graph , denoted as . In 1975, Slater was the first person to independently defined the concept of metric dimension¹. By using terminology such as resolving sets, this concept was studied a second time in². Many researchers find this area of research increasingly important because of its wide range of applications, such as robot navigation^3,4, telecommunication networks^5,6, and geographical routing protocols. Several invariants have been introduced for this term, including edge metric dimension and FT (fault-tolerant) metric dimension metric dimension. For applications of graph theory in other scientific areas, we refer to^7–10.

Starphenes are fundamental structures for the miniaturization of various electronic devices, especially organic ones. They are essential to the operation of various logical gates in the system. In¹¹, the writers examined the starphene’s resolvability parameters including the metric, the edge metric dimension, and the generalizations. Fault-tolerant and other resolvability parameter have also been studied for biswapped interconnection networks and some drug structures^12,13. It has been shown in¹⁴ that the metric dimension does not necessarily have to be a positive integer. Many infinite groups of wheel-related grids and their metric dimension have been examined in^15,16. Hayat et al.¹⁷ studied the resolvability and domination related parameters of complete multipartite graphs. Graphs of convex polytopes can be constructed using the geometric structures of convex polytopes, which can be done while preserving the adjacency-incidence relationship between the vertices. Much research has been done on these planar geometric graphs for various uses. For example, the metric dimension^18,19 and FT metric dimension on six infinite families of convex polytopes²⁰ have been investigated. Javaid et al.²⁵, examined the relationship between FT partition and metric dimension. A connection between a revolving set and an FT resolving set of some graphs was proven by Hernando et al.²⁶. Further studies on the FT metric dimension include Raza et al.²¹, Raza et al.²², Hayat et al.²³, and Siddiqui et al.²⁴.

Imagine a network in which each vertex is assigned a unique identifier by the metric generator used to generate the nodes in the network. At this point, accurate monitoring of each vertex can begin in earnest. However, if an intruder gains admittance to the network not through its vertices but rather through the links between them (edges), the intruder will not be traced until it is too late. In this case, the surveillance will not be able to live up to its commitment, and as a result, there will be a need for additional resources within the network. The authors in²⁷ attempted to locate each invader in a network in a way that could be uniquely identified, and this was done by investigating the edge metric dimension. In today’s world, various types of metric invariants can be utilized in graphs and explored in both applied and theoretical settings. This depends on how much it is being used in different settings. When defining a graph with these metric generators, many other directions are taken into consideration. However, there are still a significant number of these that are not fully examined. Moreover, the computations of these metric invariants are NP-hard^27,28. Based on these findings, the objective of this study is to describe and examine various metric invariants with the aim of contributing to our overall understanding of the subject area.

Now, we write some definitions and known results from the literature. Let be a graph with vertex set V and edge set E. The distance d between two vertices is the shortest path between them. The distance between two edges and is defined as . We say a vertex t distinguishes two vertices and (edges and ) if the distance between u and t ( and t) is different from the distance between v and t ( and t). Now, if we choose a subset from the vertex set V such that any two vertices and (any two edges and ) from the vertex set (edge set) are distinguished by at least one vertex of . Then the subset is called a metric generator/resolving set (edge metric generator/edge resolving set). The smallest cardinality of the resolving set is called the metric dimension (edge metric dimension) denoted by (). The upper metric dimension of a graph , denoted , is defined as the maximum cardinality of a minimal resolving set of .

If is a resolving set/metric generator such that removing a vertex from that set still results in a metric generator, then that type of set is called an FT metric generator. The FT metric dimension, denoted by , refers to the smallest cardinality of the resolving set.

In linear algebra, one of the properties that bases of a vector space can have is the exchange property. However, resolving sets may not have the exchange property, even though they act like vector space bases. The resolving sets of a graph have the exchange property if, whenever and are the two minimal resolving sets, there exists and such that if we remove the vertex from and add to , the resulting set is still a minimal resolving set. When a graph holds the exchange property this means every minimal resolving set has the same size, and algorithmic strategies in order to find its metric dimension become feasible. To demonstrate the absence of the exchange property in a specific graph, it suffices to identify two minimal resolving sets with distinct sizes. For further details on this topic, see^29–31.

The above definitions utilize the basic or general scenario of metric generators. It is also one of the frequently observed cases that are commonly used in ongoing research. Several scholars have developed alternate kinds of metric generators to investigate various viewpoints on metric generators from various angles. This type of structure includes resolving to dominate sets³², also known as metric-locating-dominating sets³³, independent resolving sets³⁴, local metric sets³⁵, strong resolving sets³⁶, metric generators (K-dimension graphs)³⁷, resolving partition graphs³⁸, strong resolving partitions³⁹, and other types of structures. A handful of more fascinating articles on the same subject can be found in the literature and are well worth reading.

This manuscript primarily investigates the graph family , with its metric dimension explored in⁴⁰. In the coming sections, we will investigate the distance related parameters. In particular, , and . For this investigation we choose hexagonal Möbius graphs and barycentric subdivision of Möbius ladders (see⁴¹). The exchange property has been examined as well. We also give a relationship between upper dimension and FT metric dimension. We apply a unique method to identify the FT resolving sets. The investigation of exchange property for resolving sets leads us to this method which is quite interesting.

Next, we study the exchange property of the r-dimensional hexagonal Möbius ladder graphs.

Exchange property

Next theorem shows that the r-dimensional , where does not hold the exchange property.

Theorem 2.1

In for , the exchange property is not satisfied by the minimal metric generator.

Proof

We can choose without any loss of generality. In this case, we can represent the metric basis as [see⁴⁰]. So, it is a minimal metric generator. Furthermore, is also a minimal metric generator. There is no that would allow to remain a a metric generator.

Choose , then . If , then . If , then and when , then . Since the cardinality of minimal metric generators varies, the exchange property is not maintained.

In the following, we investigate the FT metric dimension of both and its barycentric subdivision, known as .

Investigation of and

Javaid et al.²⁵ proposed the following relation for calculating the lower bound of FT metric dimension:

1

To establish an upper limit for the FT metric dimension, one may employ the following theorems.

Theorem 3.1

²⁰ Suppose is a resolving set/metric generator of graph , then is an FT resolving set of .

Theorem 3.2

.

Proof

We can choose without losing any generality. In this case, we can represent the metric basis as [see⁴⁰]. So, it is a metric genertor. , , . Moreover, . Therefore, utilizing Theorem 2, we determine that constitutes an FT resolving set of cardinality 8 for .

Table 2.

Edge codes for of .

d(., .)

Additionally, we uncover in the proof of Theorem 3, that is a metric generator. Moreover, . Therefore, employing Theorem 2, it can be determined that comprising of the elements serves as an FT resolving set for , with a cardinality of 8. The above discussion and inequality (1) shows that 8 is the upper bound and 4 is the lower bound for .

Table 3.

Edge codes for of .

d(., .)

Theorem 3.3

.

Proof

Case 1. Assuming , where , and for the sake of generality, we can proceed under the assumption that i equals 0. n this case, the set serves as a metric generator for , as demonstrated in reference⁴¹. , , . Moreover, . Therefore, applying Theorem 2, we can determine that , consisting of the elements , , forms an FT resolving set for with a total cardinality of 10.

We also find that is a metric generator [see⁴¹]. , , , . Moreover, . Utilizing Theorem 2, it can be established that , comprising the elements , also forms an FT resolving set for with a cardinality of 9.

Case 2. Assuming , where , and for the sake of generality, we can proceed under the assumption that i equals 0. In that case, the set serves as a metric basis for , as demonstrated in reference⁴¹, thus making it a metric generator as well. , , . Moreover, . Therefore, with the application of Theorem 2, it can be deduced that , comprising the elements , serves as an FT resolving set for with a total cardinality of 10.

Additionally, it can be observed that serves as a resolving set, as detailed in reference⁴¹. . Moreover, . Hence, employing Theorem 2, we can conclude that , consisting of the elements , , also serves as an FT resolving set for with a total cardinality of 9. The above discussion and inequality (1) shows that 10 is the upper bound and 4 is the lower bound for .

Case 3. Assuming r to be equal to , where , and for the sake of generality, we can proceed under the assumption that i equals 0. In that case, the set serves as a metric basis for , as demonstrated in reference⁴¹, thereby making it a metric generator as well. , , . Moreover, . Therefore, with the application of Theorem 2, it can be determined that , comprising the elements ,, forms an FT resolving set for with a total cardinality of 10.

Furthermore, it can be observed that serves as a metric generator, as detailed in reference⁴¹. , , , . Moreover, . Therefore, applying Theorem 2, we can conclude that , composed of the elements , , also serves as an FT resolving set for with a total cardinality of 9. The above discussion and inequality (1) shows that 10 is the upper bound and 4 is the lower bound for .

Case 4. Assuming r to be equal to , where , it is permissible, for the sake of generality, to make the assumption that i equals 0. In that scenario, the set acts as a metric basis for , as evidenced in reference⁴¹, consequently making it a metric generator as well. , , . Moreover, . Hence, applying Theorem 2, it can be determined that , consisting of the elements , constitutes an FT resolving set for with a cardinality of 10.

Additionally, it can be observed that serves as a resolving set, as indicated in reference⁴¹. , , , . Moreover, . Therefore, through the application of Theorem 2, it can be determined that , consisting of the elements , also functions as an FT resolving set for with a total cardinality of 9. The above discussion and inequality (1) shows that 10 is the upper bound and 4 is the lower bound for .

In the preceding proof, we established an FT resolving set with a cardinality of 9 by employing a minimal resolving set of size 4. The following problem can be suggested.

Conjecture 1

If we consider the barycentric subdivisions of Möbius ladders as , then equals 9.

Relationship between and

The metric generator with maximum cardinality is called the upper bases of that graph.⁴². Chartrand⁴² et al. also gave the following relation between and a graph as follows

2

We can establish the relationship between and using the exchange property. In the introduction, it is stated that if the exchange property is satisfied, it implies that every minimal resolving set for the graph will have an equal size. In this case , so we get

.

Remark 1

Consider a graph with k resolving sets, out of which are minimal resolving sets. Given that a minimal resolving set cannot serve as an FT resolving set, this reduces the possibility of FT resolving sets to a count of .

Investigation of

The vertex and edge set of is as follow: and . We want to show that it has a constant edge metric dimension.

Lemma 1

.

Proof

Case (i). When , we can express it as , with . The metric generator is the set selected for an index i within the range of . The codes for the edges relative to are specified as follows:

• For : .

• For : .

• For : .

• For : .

• For : .

The remaining codes are provided in Tables 1 and 2.

Table 1.

Edge codes for of .

d(., .)

Case (ii). When , we can express it as , with . The metric generator is the set selected for an index i within the range of . The edge codes relative to are as follows:

• For : .

• For : .

• For : .

• For : .

• For : .

Codes for the edges for given in Table 3 and for is given in Table 4.

Table 4.

Edge codes for of .

d(., .)
			2q

Lemma 2

.

Proof

Suppose on contrary that . We can assume, without losing any generality, that constitutes a metric generator, with . However, this leads to the result:

• When we can observe that and both take the form .

• When it can be observed that and are both equal to

• When , it is observed that and both take the form .

• When (where r is an odd number), we observe that and are both equal to .

• When (assuming r is even), and are both given by .

• When equals (with the condition that r is even), and are both equal to .

• When , and both take the form .

There is a contradiction present.

Theorem 5.1

Consider the hexagonal Möbius graph , where the vertex set is denoted as and the edge set as . In this graph, it can be established that equals 3.

Proof

By Lemmas 1, 2 we get .

Theorem 5.1 answers an open problem in⁴⁰.

Investigation of

This section is dedicated to exploring the upper limit for the mixed metric dimension of the hexagonal Mbius graph.

Theorem 6.1

when and when .

Proof

When , it can be expressed as where The metric generator is the set chosen for an index i . The tables below provide the codes for the vertices and edges in relation to (Tables 5, 6, 7, 8, 9,10, 11, 12).

Table 5.

Vertex and edge codes for and of when .

d(., .)
	0	1		2	r
	r	r	2		2
	r		3	r	1
	r	r	2	r	0
	1	2	r	3
	0	0		2	r
			0		1
	1	2	r	0	r
	1	2		1
	2	2		2
			2		1
	r		2	r	0
	1	2	r	2
	0	1		2
	0	1		1	n
	r		1		0

Table 6.

Vertex codes for of .

d(., .)

Table 7.

Edge codes for of .

d(., .)

Table 8.

Edge codes for of .

d(., .)

Table 9.

Vertex and edge codes for and of when .

d(., .)
	0		2	r
	1	r	3
		0		2
	r	1		2
	1	r	0	r
	r	1	r	0
	1	r	2
	0		2

Table 10.

Vertex codes for of .

d(., .)

Table 11.

Edge codes for of .

d(., .)

Table 12.

Edge codes for of .

d(., .)

When , this can be expressed as , where .

The metric generator is the set chosen for an index i . The following tables provide the codes for the vertices and edges concerning .

Investigation of

The investigation of the upper bound for the metric dimension of the Mbius octagonal chain has been conducted in reference⁴⁰. We will determine the lower bound of and this leads to the conclusion that it has metric dimension 3.

Lemma 3

If we define the set of vertices of as , it follows that the lower bound of is 3.

Proof

Suppose on contrary that . Let us consider, without losing any generality, that serves as a metric generator, where . However, this leads to the result:

• For , we have .

• when r is even, we have .

• For , we have .

• .

• .

• .

A contradiction.

Theorem 7.1

.

Proof

We determined that the metric dimension is precisely 3 based on the information provided in Lemma 3 and Table 4.1 from the source⁴⁰.

Resolvability parameters of lower benzenoid hydrocarbons

In⁴³, Khan studied applications of domination-related graph parameters in structure-property modeling of the total -electronic energy () of lower benzenoid hydrocarbons (BHs). In particular, the efficacy of existing domination-related graph parameters in predicting the of lower 30 BHs was investigated. Those 30 BHs have been depicted in Figure 1.

Figure 1.

The lower 30 BHs.

At the end of the study, Khan⁴³ asked to investigate other families of graph-theoretic parameters such as resolvability-related parameters. In view of that, in this paper, we compute the exact values of the metric dimension, the edge metric dimension, and the mixed metric dimension of the 30 BH graphs given in Figure 1.

Metric dimension of BH graphs

We want to show that Benzene, Naphthalene, Anthracene, Tetracene, Pentacene, and Hexacene have metric dimension 2.

Lemma 4

.

Proof

When , we can express it as , with . The metric generator is the set . The codes for the vertices relative to are specified as follows:

• For : , with .

• For : , with .

• For : , with .

• For : , with , .

• For : , with , .

• For : , with , .

• For : , with , .

• For : , with , .

• For : , with .

• For : , with , .

• For : , with , .

• For : , with .

• For : , with , .

• For : , with , .

Now, we want to show that Phenanthrene, Benzo[c]phenanthrene, Benzo[a]anthracene, Chrysene, Triphenylene, Pyrene, Benzo[a]tetracene, Dibenzo[a,h]anthracene, Dibenzo[a,j]anthracene, Pentaphene, Benzo[g]chrysene, Benzo[c]chrysene, Picene, Perylene, Benzo[b]chrysene, Dibenzo[a,c]anthracene, Dibenzo[b,g]phenanthene, Benzo[e]pyrene, Benzo[a]pyrene, Benzo[ghi]perylene, Ovalene have metric dimension 2. The corresponding vertex codes are delivered in Tables 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33.

Table 13.

Vertex codes for phenanthrene.

d(., .)

Table 14.

Vertex codes for benzo[c]phenanthrene.

d(., .)

Table 15.

Vertex codes for benzo[a]anthracene.

d(., .)

Table 16.

Vertex codes for chrysene.

d(., .)

Table 17.

Vertex codes for triphenylene.

d(., .)

Table 18.

Vertex codes for pyrene.

d(., .)

Table 19.

Vertex codes for benzo[a]tetracene.

d(., .)

Table 20.

Vertex codes for dibenzo[a,h]anthracene.

d(., .)
	3	5

Table 21.

Vertex codes for dibenzo[a,j]anthracene.

d(., .)
	0	8

Table 22.

Vertex codes for pentaphene.

d(., .)

Table 23.

Vertex codes for benzo[g]chrysene.

d(., .)
	6	1

Table 24.

Vertex codes for benzo[c]chrysene.

d(., .)

Table 25.

Vertex codes for picene.

d(., .)
	2	5

Table 26.

Vertex codes for perylene.

d(., .)

Table 27.

Vertex codes for benzo[b]chrysene.

d(., .)

Table 28.

Vertex codes for dibenzo[a,c]anthracene.

d(., .)

Table 29.

Vertex codes for dibenzo[b,g]phenanthene.

d(., .)

Table 30.

Vertex codes for benzo[e]pyrene.

d(., .)
	6	1

Table 31.

Vertex codes for benzo[a]pyrene.

d(., .)

Table 32.

Vertex codes for benzo[ghi]perylene.

d(., .)

Table 33.

Vertex codes for ovalene.

d(., .)

Lemma 5

.

Proof

The codes for the vertices relative to are specified as follows:

Now, we want to show that Pentahelicene, Hexahelicene and Coronene have metric dimension 3. The corresponding vertex codes are delivered in Tables 34, 35.

Table 34.

Vertex codes for pentahelicene and hexahelicene.

d(., .)
	0	4	6
	1		7
	2
	3
	2	4	6
	3	1	5
	1	3	5

Table 35.

Vertex codes for coronene.

d(., .)

Lemma 6

.

Proof

The codes for the vertices relative to are specified as follows:

Edge metric dimension of BH graphs

We want to show that Benzene, Naphthalene, Anthracene, Tetracene, Pentacene, and Hexacene have edge metric dimension 2.

Lemma 7

.

Proof

When , we can express it as , with . The metric generator is the set . The codes for the vertices relative to are specified as follows:

• For : , with .

• For : , with .

• For : , with .

• For : , with .

• For : , with .

• For : , with .

• For : , with .

• For : , with .

• For : , with .

• For : , with .

• For : , with .

• For : , with .

• For : , with .

• For : , with .

• For : , with .

Now, we want to show that Phenanthrene, Benzo[c]phenanthrene, Benzo[a]anthracene, Chrysene, Triphenylene, Pyrene, Benzo[a]tetracene, Dibenzo[a,h]anthracene, Dibenzo[a,j]anthracene, Pentaphene, Benzo[g]chrysene, Benzo[c]chrysene, Picene, Perylene, Benzo[b]chrysene, Dibenzo[a,c]anthracene, Dibenzo[b,g]phenanthene, Benzo[e]pyrene, Benzo[a]pyrene, Benzo[ghi]perylene, Ovalene have edge metric dimension 2. The corresponding edge codes are given in Tables 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56.

Table 36.

Edge codes for phenanthrene.

d(., .)
	2
	3	2
		3

Table 37.

Edge codes for benzo[c]phenanthrene.

d(., .)
		5
	1
	0
		0
	6
		2
	4	2

Table 38.

Edge codes for benzo[a]anthracene.

d(., .)
	2
	7	5
		2

Table 39.

Edge codes for chrysene.

d(., .)
	0
		4
	4
	7
		1
		0
	5	1

Table 40.

Edge codes for triphenylene.

d(., .)
	5
	4
		4
	6
		4
		5

Table 41.

Edge codes for pyrene.

d(., .)
	0
		4
	2
	2	1
	4

Table 42.

Edge codes for benzo[a]tetracene.

d(., .)
	2
		4
	7
	8
	6	2

Table 43.

Edge codes for dibenzo[a,h]anthracene.

d(., .)
		6
		4
	4
		5

Table 44.

Edge codes for dibenzo[a,j]anthracene.

d(., .)
		8
		7
	4
		4
	7
	8

Table 45.

Edge codes for pentaphene.

d(., .)
	2
		2
		3
	7	4
		5

Table 46.

Edge codes for benzo[g]chrysene.

d(., .)
	7
	4
		6
		4
		5
		0
		1

Table 47.

Edge codes for benzo[c]chrysene.

d(., .)
	8
	2
	3

Table 48.

Edge codes for picene.

d(., .)
	3
		3
		2
	4
		4
		5
	5	1

Table 49.

Edge codes for perylene.

d(., .)
	0
		5
	3
	4
		4
	5
		3
		4
	4

Table 50.

Edge codes for benzo[b]chrysene.

d(., .)
		7
		3
		2
	6
		4

Table 51.

Edge codes for dibenzo[a,c]anthracene.

d(., .)
	2
	7
	6	4
	5
	6

Table 52.

Edge codes for dibenzo[b,g]phenanthene.

d(., .)
	5
		4
	2

Table 53.

Edge codes for benzo[e]pyrene.

d(., .)
	0
		5
		4
		6
	6
		3
	5
		0
		4
		2
	4	3

Table 54.

Edge codes for benzo[a]pyrene.

d(., .)
		5
		6
		4
		3
	5	1
		2

Table 55.

Edge codes for benzo[ghi]perylene.

d(., .)
		5
		6
	6
	5
		4

Table 56.

Edge codes for ovalene.

d(., .)
	0	5
		8
	5
		5
	8
		3
	2
		6
	4
		2
		4

Lemma 8

.

Proof

The codes for the edges relative to are specified as follows:

Now, we want to show that Pentahelicene, Hexahelicene and Coronene have edge metric dimension 3. The corresponding edge codes are given in Tables 57, 58, 59.

Table 57.

Edge codes for pentahelicene.

d(., .)
			6
	2	1	5
		0	5
	4
	5
	6	6
	7	7

Table 58.

Edge codes for hexahelicene.

d(., .)
	0	4	6
		5	7
	2		6
	3
		0	5
	4
	5
	6	6
	7	7
			2
		2
		3	5

Table 59.

Edge codes for coronene.

d(., .)
	0	0	6
			5
	6	5
	6	6	0
	5	6
			6
	3	4	4
		2
	4		2

Lemma 9

.

Proof

The codes for the edges relative to are specified as follows:

Mixed metric dimension of BH graphs

We want to show that Benzene, Naphthalene, Anthracene, Tetracene, Pentacene, and Hexacene have mixed metric dimension 3.

Lemma 10

.

Proof

When , we can express it as , with . The metric generator is the set . The codes for the vertices relative to are specified as follows:

• For : , with .

• For : , with .

• For : , with .

• For : , with .

• For : , with .

• For : , with .

• For : , with .

• For : , with .

• For : , with .

• For : , with .

• For : , with .

• For : , with .

• For : , with .

• For : , with .

• For : , with .

• For : , with .

• For : , with .

• For : , with .

• For : , with .

• For : , with .

• For : , with .

• For : , with .

• For : , with .

• For : , with .

• For : , with .

• For : , with .

Now, we want to show that Phenanthrene, Benzo[a]anthracene, Triphenylene, Pyrene, Benzo[a]tetracene, Pentaphene, Dibenzo[a,c]anthracene, Benzo[e]pyrene, Benzo[a]pyrene, Benzo[ghi]perylene, Coronene, Ovalene have mixed metric dimension 3. The corresponding mixed codes are delivered in Tables 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71.

Table 60.

Mixed codes for phenanthrene.

d(., .)
			3
			4
			2
	6
			0
		2	1
			2

Table 61.

Mixed codes for benzo[a]anthracene.

d(., .)
	0	7
		6

Table 62.

Mixed codes for triphenylene.

d(., .)
			4
		0	5
		6
	4		2
		4
		5

Table 63.

Mixed codes for pyrene.

d(., .)
			6
	2	1
		0
	4
	5		0
		4
		3
	3		2
		2

Table 64.

Mixed codes for benzo[a]tetracene.

d(., .)
	0	7
	8
	9		0
	7

Table 65.

Mixed codes for pentaphene.

d(., .)
	0	9
		0	5
		6
	2	7	4
	8	2	3

Table 66.

Mixed codes for dibenzo[a,c]anthracene.

d(., .)
	0	7
		0	5
	6
	7		0

Table 67.

Mixed codes for benzo[e]pyrene.

d(., .)
	0	5
	6
		4	1
	5		0
			4
		2	3
			2

Table 68.

Mixed codes for benzo[a]pyrene.

d(., .)
	0	7
			4
	6	1	3
		0	3
		4
			3
			4
	6
			2

Table 69.

Mixed codes for benzo[ghi]perylene.

d(., .)
	0	5
			6
		0	5
	6
	5		0
		4
			4
	4

Table 70.

Mixed codes for coronene.

d(., .)
			6
		1	5
			6
			5
	6
	5
		6
		5
	2
		2
			2

Table 71.

Mixed codes for ovalene.

d(., .)
			8
		1	7
			8
	4	1	7
		0	7
	6
	7
		8
			7
			6
	4
		4
	5		2

Lemma 11

.

Proof

The codes for the vertices and edges relative to are specified as follows:

Now, we want to show that Benzo[c]phenanthrene, Chrysene, Dibenzo[a,h]anthracene, Dibenzo[a,j]anthracene, Benzo[g]chrysene, Pentahelicene, Benzo[c]chrysene, Picene, Perylene, Benzo[b]chrysene, Dibenzo[b,g]phenanthene have mixed metric dimension 4. The corresponding mixed codes are given in Tables 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82.

Table 72.

Mixed codes for benzo[c]phenanthrene.

d(., .)
	0	1	5	7
		0	5	7
				6
	4	4	1	5
			0
	5	6
	6	7		0
	4	5	2	2

Table 73.

Mixed codes for chrysene.

d(., .)
	2
	3
			0
	6	8		0
			2
			3
		2	4	6

Table 74.

Mixed codes for dibenzo[a,h]anthracene.

d(., .)
	1	2	5	9
		0	6	10
	2
	9	10	4	0
			5

Table 75.

Mixed codes for dibenzo[a,j]anthracene.

d(., .)
	8	9	1	3
		0	7	9
				8
	4
	5	6	1	5
			0
	7	8
	8	9		0

Table 76.

Mixed codes for benzo[g]chrysene.

d(., .)
	0	1	6	8
	1	2	5	7
		0	6	8
	2
				4
	5	6	0	5
	6	7
	5	6
	6	7	4	1
	7	8		0
	3	4	4	4
			5

Table 77.

Mixed codes for pentahelicene.

d(., .)
		0	5	7
	4			6
	5
	6	6
	7	7		0

Table 78.

Mixed codes for benzo[c]chrysene.

d(., .)
			8	8
	2
	6	6	1	2
			0	3
	7	7		0
		6	4
			5
		2

Table 79.

Mixed codes for picene.

d(., .)
		1	5	9
		0	6	10
	2	2	4	8
	3
			0
			4
			2
			3
		2	4	8
	6	8		2

Table 80.

Mixed codes for perylene.

d(., .)
	3	5	2	2
			5
		1
		0	4	6
	3
			0	3
		6
	4	6		0
			4
			2
	2
	3

Table 81.

Mixed codes for benzo[b]chrysene.

d(., .)
			6	10
	2
	9	9		0
		6	2
			3

Table 82.

Mixed codes for dibenzo[b,g]phenanthene.

d(., .)
	0	1	6	8
		0	7	9
	2
				6
	5
	6	8
	7	9		0

Lemma 12

.

Proof

The codes for the vertices and edges relative to are specified as follows:

Now, we want to show that Hexahelicene has mixed metric dimension 5. See Table 83 for the mixed codes.

Table 83.

Mixed codes for hexahelicene.

d(., .)
	1	3	3	5	7
	0	1	4	6	8
		0	5	7	9
	2			6	8
	3
	3			5	7
	4	6
					6
	5	7
	6	8	6
	7	9	7		0
			2
		2	3	5	7

Lemma 13

.

Proof

The codes for the vertices and edges relative to are specified as follows:

Conclusion

In this paper, we study various resolvability related graphical parameters for Moöbius-type geometric graphs. The motivation of the problem comes from the thesis⁴⁰ by Bisharat who studied these parameters for hexagonal & octagonal Moöbius ladders and other related families of graphs. Certain open problems are raised in the conclusion section of⁴⁰. This paper solves those open problems such as finding the exact value of the edge metric dimension of n-dimensional hexagonal Moöbius ladder graphs. Additionally, we studied exchange property and the FT metric dimension of certain infinite families of Moöbius-type geometric graphs. In response of an open question by Khan⁴³, we find exact values of the metric dimension, the edge metric dimension, and the mixed metric dimension of the 30 lower benzenoid hydrocarbons. These results help in modeling the total -electronic energy of lower benzenoid hydrocarbons by means of resolvability-based graph-theoretic parameters.

In light of this research, we present the following unresolved questions:

Problem 1

Is there a connection between the and when the minimal resolving sets have varying sizes?

Problem 2

Let be the barycentric subdivisions of Mbius ladders, then is it true that ?

Problem 3

Find exact values of and .

Author contributions

Conceptualization, S.H. and R.N.; methodology, S.H., R.N. and Z.K.; software, M.B.B. and H.M.A.S.; validation, S.H., M.B.B., H.M.A.S. and Z.K.; formal analysis, A.R., R.N. and M.B.B.; investigation, S.H.; resources, A.R., M.B.B.; data curation, S.H., A.R. and Z.K.; writing–original draft preparation, H.M.A.S., S.H. and M.I.; writing–review and editing, A.R., M.B.B. and Z.K.; visualization, M.B.B. and M.I.; supervision, M.I. and S.H.; project administration, M.I. and A.R.; funding acquisition, M.B.B. and Z.K. All authors have read and agreed to the published version of the manuscript.

Funding

S. Hayat is supported by UBD Faculty Research Grants with Grant Number UBD/RSCH/1.4/FICBF/2025/011. Z. Klai extends her appreciation to the Deanship of Scientific Research at Northern Border University, Arar, KSA for funding this research work through the project number "NBU-FFR-2025-2942-03".

Data availability

The datasets used and/or analyzed during the current study available from the corresponding author on reasonable request.

Declarations

Competing interests

The authors declare no competing interests.