To understand the abstract structure of graphs, graph invariants can be a powerful tool. Metric generators in graphs have evolved into a variety of distinct types based on their popularity or usefulness. As an example, virus propagation in complicated network difficulties can be detected using the double metric dimension (DMD).

We consider a finite, undirected, and connected graph Γ with the edge set EΓ and the vertex set VΓ. In a connected graph Γ, the distance between a pair of distinct vertices, say k and l in Γ, is the smallest path length among the lengths of all paths between them and is represented by d(k, l). For any two distinct vertices j and k in VΓ, a vertex l∈VΓ resolves the vertices j and k if the condition d(j,l)≠d(k,l) holds true. Let k be a vertex of Γ and HΓ={hμ|1≤μ≤ρ} be an ordered subset of the vertex set VΓ, then r(k,HΓ) is the representation of k with respect to HΓ which is the ρ-vector (d(k,hμ))μ=1ρ, also known as vector of metric coordinates. If each vertex of graph Γ possesses a unique vector of metric coordinates with respect to HΓ then the subset HΓ is called a resolving set for graph Γ. If the subset HΓ is the resolving set with the minimum number of elements in Γ, then its count becomes the metric dimension (MD) of Γ and can be denoted as dim⁡(Γ).

It is common in computer science, chemistry, biology, and operations research to structure graph theory models. In graph theory, the computation of the resolving sets and MD for general graphs is complex. In various disciplines, such as optimization, pattern recognition, and loran detecting devices, the MD has gained all the interest due to its applications.

Slater [1], a mathematician, was the first to discover the term MD. This notion was used to describe the study of the Loran or sonar model station. Another mathematician, Harary, independently described the concept of MD with the help of Harary et al. [2]. Additionally, this concept cleared the path for finding the unique recipient of a message on a network. Since then, several studies have been done on resolving sets, including [3,4]. Many different tasks can be accomplished with resolving sets, like network discovery and verification [5], strategies for the mastermind games [6], as well as digital geometry, pattern recognition, and processing of images [7]. The minimum order of a resolving set of Hamming graphs closely relates to the problem of weighing the coins discussed in [8,9]. Also, there are numerous applications in different fields, such as chemical structures [10], robotics [11], combinatorial optimization [12], geographic routing protocols [13] for the theoretical study of the MD.

The MD of any graph is a computationally difficult problem to solve. So, valuable bounds for different types of graphs have been found. The MD of all connected graphs, as well as the MD of certain well-known graph families, such as trees, pathways, and complete graphs, were established by Chartrand et al. [10,14]. For some families of generalized Petersen graphs, MD bounds were studied in [15]. The MD for chorded cycles and kayak paddles graphs was found to be constant by Ahmad et al. [16]. Many authors have extensively studied the MD of path-related graphs. The MD of the Kenser graph was computed by Hui et al. [17]. On the other hand, Alholi et al. [18] also contributed to this area by finding that the MD is constant. Buczkowski and others [19] presented a generic concept of k-dimensional graphs. The NP-completeness of the MD for general graphs was demonstrated by Khuller et al. [11]. The minimum order of a resolving set of Jahangir graphs has also been discussed in detail by Tomescu et al. [20].

Caceres et al. [21] developed the concept of doubly resolving sets (DRSs) of a graph Γ, demonstrating that the MD of the Cartesian product Γ◻Γ is closely linked to the minimal doubly resolving sets (MDRSs) of Γ. Consider the vertices k1,k2,l1,l2∈VΓ, then any pair of vertices k1,k2 is said to be doubly resolved by the vertices l1,l2 if the condition d(k1,l1)−d(k1,l2)≠d(k2,l1)−d(k2,l2), holds true. A subset NΓ∈VΓ is said to be a DRS, if any two distinct vertices of graph Γ are doubly resolved by some two vertices of the subset NΓ. A MDRS is a DRS having minimum order. The order of MDRS is denoted by ψ(Γ) and is termed the DMD of Γ. Note that if the vertices l1,l2 doubly resolve the vertices k1,k2 then d(k1,l1)−d(k2,l1)≠0 or d(k1,l2)−d(k2,l2)≠0. For all graphs Γ, any DRS is obviously a resolving set, with dim⁡(Γ)≤ψ(Γ). The MDRS problem has been proven to be NP-hard in the context of general graphs [22]. Furthermore, it was shown in [21] that the upper bound on the MD of the Cartesian product of Γ and Ω can be stated as; dim⁡(Γ◻Ω)≤dim⁡(Γ)+ψ(Ω)−1.

Therefore, DRSs are essential for studying the MD of Cartesian product graphs. These results have inspired us to work on finding the MDRSs. Also, MDRSs themselves have a unique combinatorial nature that can be seen in their integer programming model, which was shown in [22]. There are several families of graphs for which the problem of finding the MDRS has been examined. For example, MDRSs of prisms [23], convex polytopes [24], and Hamming graphs [25] have all been studied in the literature. The family of circulant graphs in [26] was found to have the same MD and MDRS. The MDRSs for the line graphs of prisms and sunlet graphs were found in [27]. For the first time, Chen et al. explicitly approximated the upper and lower bounds for the MDRS problem [28]. In [29], the minimal resolving sets and DMD of the line graph of chorded cycles were examined. Authors demonstrated that the DMD of L(Cnt) is exactly one greater than its MD. Ahmad et al. discussed the problem of finding the minimal resolving set and MDRS for line graphs of kayak paddles graphs [30]. The authors in [31,32] presented some families of convex polytopes with constant DMD. While solving the MDRS problem, certain families of Harary graphs, layer-sun graphs have also been investigated in [33,34], respectively. Cocktail, jellyfish and necklace graphs and their line graphs have also been studied for the MD and MDRS problems, see [35–37].

Our research is the continuation of the literature work mentioned earlier. We have investigated the DMD for some particular classes of Toeplitz graphs to contribute to our knowledge of this distance-based parameter in graphs. However, this variant is helpful in many fields, for instance, diffusion over the network, epidemics in human beings, the origin of a disease outbreak, etc. Using MDRSs, it is possible to identify the source of diffusion in complicated networks. Attempting to find the source of an infection in an extensive network might be difficult, but it is still an exciting challenge. For example, to detect the spread of the virus throughout a network, we need to know when specific nodes in the network are infected. However, maintaining observer nodes that can report their infection time may be costly. In addition, the position of a node in the network affects how much information it contains. Which nodes should we use as observers in order to increase our chances of correctly identifying the source? Because of its close relationship to another well-known DRS problem, we can reduce the number of observers required to achieve perfect detection even if the initial time at which an epidemic spreads is unknown [38,39].

Liu et al. [40] determined the MD of some families of Toeplitz graph Tn(1,t) for n ≥ t + 2 given in the following theorems:

Detailed discussion of the rest of the article is provided in the following sections:

Our discussion of Toeplitz graphs in Section 2 included the computation of MDRSs for the family of Toeplitz graphs Tn(1,2) for n ≥ 4.

For a graph Γ with n vertices labelled as {1, 2, …, n} its adjacency matrix A is n × n matrix whose jkth entry is 1 if the vertex j and vertex k joined by an edge and 0 otherwise. An n × n matrix B=bjk is known as Toeplitz matrix if bjk=bj+1,k+1 for each {j, k = 1, …, n − 1}. A simple undirected graph Tn is Toeplitz graph if matrix n × n which is B=bjk is the symmetric Toeplitz matrix and for all j, k = {1, …, n} satisfied the following: edge j, k is in EΓ iff bjk=bkj=1. An n × n matrix B will be labelled 0, 1, …, n − 1 which has n distinct diagonals. The main diagonal has bjj=0 for all j = 1, …, n, so Toeplitz graph has no loop. The diagonals {x1,x2,…,xq} containing ones 0<x1<x2<…<xq<n. The Toeplitz graph Tn<x1,x2,…,xq> with vertex set {1, 2, …, n} has edge j, k for 1 ≤ j ≤ k ≤ n, occurs iff k−j=xp for some p, 1 ≤ p ≤ q. Toeplitz graphs are defined as graphs formed from Toeplitz matrices. Specifically, in this part, we computed the MDRSs for the family of Toeplitz graph Tn(1,2). Fig. 1 illustrates the graph T11(1,2).

It follows from Theorem 1.1 that ψ(Tn(1,2))≥2, for n ≥ 4. We will also calculate that ψ(Tn(1,2))=2, for n ≥ 4. To compute the distances between the vertices of the Toeplitz graph Tn(1,2): Define Sω(e0)={e∈Tn(1,2):d(e0,e)=ω}, which is the vertex subset in VTn(1,2) at distance ω from e0. The Table 1 shows the sets Sω(e0) for Toeplitz graph Tn(1,2), where n ≥ 4.

Due to the symmetry of Tn(1,2), where n ≥ 4: d(eω,eν)=d(e0,e|ν−ω|) if 0≤|ν−ω|≤n−1.

Due to the fact that, in order to calculate the distance between any pair of vertices in VTn(1,2), we must know the distance d(e0,e) for each e∈Tn(1,2).

The main theorem is stated below by using Lemmas 2.1 and 2.2.

Specifically, in this part, we computed the MDRSs for the family of Toeplitz graph Tn(1,4). Fig. 2 illustrates the graph T11(1,4).

It follows from Theorem 1.2 that ψ(Tn(1,4))≥2, for n ≥ 6. We will also calculate that ψ(Tn(1,4))=3, for n ≥ 6. To compute the distances between the vertices of the Toeplitz graph Tn(1,4): Define Sω(e0)={e∈Tn(1,4):d(e0,e)=ω}, which is the vertex subset in VTn(1,4) at distance ω from e0. The Table 4 shows the sets Sω(e0) for Toeplitz graph Tn(1,4), where n ≥ 6.

Due to the symmetry of Tn(1,4), where n ≥ 6: d(eω,eν)=d(e0,e|ν−ω|) if 0≤|ν−ω|≤n−1. Due to the fact that, in order to calculate the distance between any pair of vertices in VTn(1,4), we must know the distance d(e0,e) for each e∈Tn(1,4).

The main theorem is stated below by using Lemmas 3.2–3.5.

Specifically, in this part, we computed the MDRSs for the family of Toeplitz graph Tn(1,3). Fig. 3 illustrates the graph T11(1,3).

It follows from Theorem 1.3 that ψ(Tn(1,3))≥3, for n ≥ 5. We will also calculate that  ψ(Tn(1,3))={3,for 5≤n≤7;4,for n≥8.

To compute the distances between the vertices of the Toeplitz graph Tn(1,3): Define Sω(e0)={e∈Tn(1,3):d(e0,e)=ω}, which is the vertex subset in VTn(1,3) at distance ω from e0. The Table 10 shows the sets Sω(e0) for Toeplitz graph Tn(1,3), where n ≥ 5.

Due to the symmetry of Tn(1,3), where n ≥ 5: d(eω,eν)=d(e0,e|ν−ω|) if 0≤|ν−ω|≤n−1. Due to the fact that, in order to calculate the distance between any pair of vertices in VTn(1,3), we must know the distance d(e0,e) for each e∈Tn(1,3).