The distance between two vertices

The metric dimension is used in a variety of scientific disciplines, including chemical graph theory and computer networking. A technique for finding a vertex’s precise location or position in a network is called localization. In a work environment, localization is used when a computer sends a printing command to help locate nearby printers, broken equipment, network intruders, illegal or unauthorised connections, and wandering robots. The localization of a network is a difficult, expensive, time-consuming, and arduous procedure. The minimum number of vertices (the metric dimension of a graph representing the network) is picked in such a way that, with the aid of selected vertices, the location of the required vertex may be identified by its distinctive representation.

In robotic engineering, there is no concept of direction and no visibility on a polygonal type planar surface/mesh. So handling the navigation of a robot (a navigation agent) from point to point is a crucial task, which can be done quickly with the help of distinctively labelled landmarks. These landmarks help the robot locate itself on the surface (or graph). The visual detection of a landmark sends information to the robot about its direction, allowing it to determine its position. In this way, the robot’s position on the graph can be determined by its distances to the elements of the set of distinctively labelled landmarks. The problem of locating the fewest number of landmarks to determine the robot’s position is equivalent to finding a minimum metric generator of the graph on which the robot’s navigation is required [

Consider a connected graph

A vertex

The set

Slater introduced the concept of metric identification by using the concept of metric generator with name reference (locating) set [

Some fundamental problems related to the metric identification in tree graphs and graphs having minimum and maximum metric dimension have been addressed in [

Using an algorithmic technique with mathematical induction, the problem of metric identification has been solved for a family of 3-regular circulant graphs by Salman et al. [

For three families,

The study of metric identification has also been taken into account for various chemical networks such as for chordal ring networks in [

Various graph products have also been considered in the context of metric identification problem such as the lexicographic product in [

The following theorem provides the minimum metric dimension for a connected graph:

A connected graph in which no edge is a part of more than one cycle is called a cactus graph, see

Let

A chain polygonal cactus, denoted by

According to the definition, cactus chain

In

We can see that all the metric vectors are distinct. So,

Obviously for every two vertices

Whenever

Whenever

Hence, our supposition is wrong and

Now, we prove that

If

If

When

If

If

If

According to all these cases, it can be concluded that

Neighbors

We suppose that

Both the claims provide that our assumption is wrong. Hence

When

When

Hence,

Whenever

Whenever

Without loss of generality, we let

With the similar justification proposed for the proof of Lemma 2.2, we have the following result:

It can be easily verified that for every pair

Suppose contrarily that

Whenever

Whenever

Thus, according to these possibilities,

Let us consider a set

It can be seen that all the metric vectors are different, which implies that

Let

Accordingly, we have the following remark:

In

Let

for fixed

This implies that

Whenever

Whenever

Hence

It follows that our supposition is wrong, and no 2-element set is a metric generator for

It can easily verify that all metric vectors are distinct. Thus,

Now, we claim that if

If

If

Whenever

Whenever

All these possibilities conclude that our supposition is wrong and

For

It can easily verify that for each pair of distinct vertices (

According to the similar reasoning of the proofs of Theorems 2.4 and 2.5 we have the following two results for

Now, let

For

It can easily verify that for every two distinct vertices

A star polygonal cactus is a

We have the following results on metric dimension problem regarding star cactus.

For

For

For

It can be seen that all the metric vectors are distinct, which yields that

From the above Lemma, we have the following consequence:

If

If

If

So, the pair

If

All these cases proved that

A family of graphs has a constant metric dimension if

The family

there were only three polygons in

there were only four polygons in

otherwise,

The family

otherwise,