The aim of robot navigation functionality is to attain the coveted position promptly whenever it is desired. Let us imagine that robot navigation in a sensor network that can obtain the distances to a collection of landmarks. A robot's position is solely resolved by determining the subset of nodes in the sensor network. It can be achieved by the concept of landmarks in the graphs introduced in Khuller et al. [1]; this idea was later named the metric dimension. All the graphs considered here have no loops and are simple, measurable, and undirected.

Let Γ = (V, E) be the graph of the distance d_Γ(a, b) (or d(a, b)) among the vertices a, b ∈ V(Γ) the minimum length of paths between them. For a vertex a ∈ V, distinguish two vertices in a graph, say b and c, if the condition d_Γ(a, b) ≠ d_Γ(a, c) holds. A set R ⊂ V(Γ) is the metric generator if some chosen vertices of the set R recognizes a pair of distinguished vertices. The metric basis with the least number of elements is called the metric generator, and the cardinality of its metric basis is termed the metric dimension. The notation employed here is β(Γ). The fundamental concept of the metric dimension was instated by Slater [2], and the notation of the metric dimension was initiated by Haray and Melter [3]. This concept was later studied by many researchers with unique modifications; for reference, see [4–8]. Some of the recent results on metric dimension and its further variations are studied in Shao et al. [9] and Raza et al. [10–13].

Lemma 1. Suppose R is the distinguishing set of Γ and the vertices a, b ∈ V(Γ). If d_Γ(a, c) ≠ d_Γ(a, c), for all vertices c ∈ V(Γ)\{a, b}, then {a, b}⋂R ≠ ∅.

Analogous to this definition, Kelenc et al. [14] introduced the concept of edge metric dimension, and this was further studied in Zubrilina [15], Peterin and Yero [16], and Zhu et al. [17]. This distance between an edge e = ab and a vertex c is given as follows

A vertex c ∈ V(Γ) distinguishes two edges of a graph e₁, e₂ ∈ E(Γ) if d_Γ(e₁, u) ≠ d_Γ(e₂, u). The set R_e ⊂ V is termed as the edge metric generator if some distinct edges of Γ are distinguished by the vertex set R_e. The cardinality of an edge metric generator is called an edge metric dimension, and it is depicted as β_e(Γ). Having defined the notion of an edge metric generator, which distinctly recognizes every edge in a graph, a common assumption would be that any edge metric generator R_e would be a metric dimension as well. This assumption is far from reality, as indicated in Kelenc et al. [14], but there are several families of graphs where this occurs, that is, β(Γ) = β_e(Γ). Some other distance related parameters are studied in Liu et al. [18–22].

In this paper, our focus is on mixed metric dimension, which is a mixed version of metric and edge metric dimension. A set R_m of vertices of a graph Γ is known as a mixed metric generator if any two distinct elements (vertices or edges) of a graph are recognized by some the vertex set of R_m. The least cardinality of a mixed metric generator for a graph is termed as a mixed metric dimension, denoted as β_m(Γ). The idea is recently brought forward by Kelenc et al. [23].

Lemma 2. Let for some vertex a ∈ V(Γ), and let R_m = V(Γ)\a, and if there is an element b ∈ N(a), also for some c ∈ R_m, d_Γ(ab, c) ≠ d_Γ(b, c), then R_m is the mixed metric generator for the graph Γ.

The notion of a mixed metric dimension clearly indicates that a mixed metric generator is also a standard metric generator and an edge metric generator, The following relationship is given in [23],

The following remark shows the structure of mixed metric dimension:

Remark 1: [23] Suppose for some graph Γ we have 2 ≤ β_m ≤ n. Recently, this concept has attracted some attention, and it has been studied by Raza et al. [24]. The authors discussed the mixed metric dimension among the prism graphs, which are commonly known as generalizes Petersen graphs P(n, 1), and two families of convex polytopes A_n, R_n, further presenting the problem of finding β_m(P(n, 2)).

Some of the results regarding metric and edge metric dimension are given:

Remark 2: [14] For n ≥ 2, the metric and edge metric dimension are, β(Pn)=βe(Pn)=1; for cycle graph, β(Cn)=βe(Cn)=2; for complete graph, β(Kn)=βe(Kn)=n-1; and for any complete bipartite graph (Kr,t) different from (K1,1), β(Kr,t)=βe(Kr,t)=r+t-2.

The generalized Petersen graphs is introduced by Watkins [25]. The P(n, ℓ), where n ≥ 3 and 1≤ℓ≤⌊n-12⌋ (see Figure 2), which is the cubic graph consists of vertices and edges, is shown below.

Example: We used the graph of P(n, 8), as can be seen in Figure 1. The mixed metric generator for P(n, 8) is β_m = {q₀, q₁, p₄, p₅}, and it can been seen from Table 1 that all the representation of vertices and edges are distinct.

The graph of the generalized Petersen graph comprises of three types of edges, external edges, internal edges, and spokes between q_i and q_i+1, p_i and p_i+m, and q_i and p_i, respectively. The vertices q_i and p_i (0 ≤ i ≤ n − 1) are termed as external and internal vertices, respectively.

The prism graph D_n is known as P(n, 1) for m = 1. Some of the already known results are given as

Theorem 1. [26] The metric dimension of Dn, for n ≥ 4:

Theorem 2. [27] When, n ≥ 4, βe(Dn)=3.

Theorem 3. [24] For n ≥ 5,

The known results for P(n, 2) concerning metric and an edge metric dimension are

Theorem 4. [28] For n ≥ 5, the metric dimension is β(P(n, 2)) = 3.

Theorem 5. [27] For n ≥ 5, β_e(P(n, 2)) = 3.

It is quite natural to investigate the mixed metric dimension of P(n, 2). Now, we will find mixed the metric dimension of (P(n, 2)), and for this the following lemmas are presented.

Lemma 4. Case 1: If n ≡ 0(mod)4, then β_m(P(n, 2)) ≤ 4.

Proof: The proof is n = 4r, r ≥ 4, where r ∈ ℤ⁺. The distinguishing vertices that will distinguish the whole vertices and edges of the graph are R_m = {q₀, q₁, p_2r, p_2r+1}. The following representations are presented with respect to R_m.

Representation of external vertices:

and,

Representation of internal vertices:

and,

Representation of external edges:

and,

Representation of external and internal edges:

and,

Representation of internal edges:

and,

     □

Case 2: For n ≡ 2(mod)4, we have β_m(P(n, 2) ≤ 4

Proof: Now we can see n = 4r + 2, r ≥ 4, where r ∈ ℤ⁺. The set of vertices that will distinguish the whole vertices and edges of the graph are R_m = {q₀, q₁, p_2r+1, p_2r+2}. The following representations are presented with respect to R_m.

Representation of external vertices:

and,

Representation of internal vertices:

and,

Representation of external edges:

and,

Representation of external and internal edges:

and,

Representation of internal edges:

and,

     □

Now from lemma3, the resolving set R_m contains vertices from external and internal cycles; that is, the resolving set cannot comprise either external or internal vertices.

Lemma 5. When n ≡ 0, 2(mod4), then β_m(P(n, 2)) ≥ 4.

Proof: Suppose that β_m(P(n, 2)) = 3, the following contradictions arises:

Similarly, other contradictions can be assumed; all the cases mentioned above suggest that β_m(P(n, 2)) ≥ 4, which clearly indicates that β_m(P(n, 2)) = 4 for n ≡ 0(mod4). Similar kind of contradictions can be proved for n ≡ 2(mod4).

     □

Remark 3: From the above cases, it can be deduced that if the mixed metric generator R_m for P(n, 2) contains two vertices of one cycle, then R_m contain at least two vertices of another cycle.

Lemma 6. β_m(P(n, 2) ≤ 5, for n ≡ 1(mod)4

Proof: Case 1: Now we can write, if n = 4r + 1, r ≥ 4, where r ∈ ℤ⁺. The set of vertices that will distinguish the whole vertices and the edges of the graph are R_m = {q₀, q₁, p₁, p_2r+1, p_2r+2}. The following representations are presented with respect to R_m.

Representation of external vertices:

and,

Representation of internal vertices:

and,

Representation of external edges:

and,

Representation of external and internal edges:

and,

Representation of internal edges:

and,

     □

Proof: Case 2: Now we can write, if n = 4r + 3, r ≥ 4, where r ∈ ℤ⁺. The set of vertices which will distinguish the whole vertices, and edges of graph are R_m = {q₀, q₁, p₁, p_2r+3, p_2r+4}. The following representations are presented with respect to R_m.

Representation of external vertices:

and,

Representation of internal vertices: