The Metric Dimension of Subdivisions of Lilly Graph, Tadpole Graph and Special Trees

: Consider a robot that is navigating in a space represented by a graph and wants to know its current location. It can send a signal to find out how far it is from each set of fixed landmarks. We study the problem of computing the minimum number of landmarks required, and where they should be placed so that the robot can always determine its location. The set of nodes where the landmarks are located is called the metric basis of the graph, and the number of landmarks is called the metric dimension of the graph. On the other hand, the metric dimension of a graph G is the smallest size of a set B of vertices that can distinguish each vertex pair of G by the shortest-path distance to some vertex in B . The finding of the metric dimension of an arbitrary graph is an NP-complete problem. Also, the metric dimension has several applications in different areas, such as geographical routing protocols, network discovery and verification, pattern recognition, image processing, and combinatorial optimization. In this paper, we study the metric dimension of subdivisions of several graphs, including the Lilly graph, the Tadpole graph, and the special trees star tree, bistar tree, and coconut tree.


Introduction
Let G=(V, E) be a connected graph and d(u,v) be the shortest path between two vertices u,v ∈ V(G).An ordered vertex set B={x 1 ,x 2 ,...,x k }⊆ V(G) is a resolving set of G if the representation r(v| B) = (d(v, x 1 ), d(v, x 2 ),....,d(v, x k )) is unique for every v∈V(G).The metric dimension of G, denoted dim(G), is the cardinality of minimum resolving set of G.
Slater [1,2] introduced the notion of a minimum resolving set as a locating set of G and uses the cardinality of B as a locating number to uniquely identify the location of an intruder in a network.Harary and Melter in [ 3] introduced independently the notion of minimum resolving set as a metric basis of G and the cardinality of B as the metric dimension of G and since then it has been used in several applications such as robot navigation in networks [4,5], application to pharmaceutical chemistry Chartrand et al. [6], application to pattern recognition Melter et al. [ 7], and application to wireless sensor network localization [8].
Despite Khuller et al [4] have shown that the problem of determining the metric dimension of a graph is NP-complete, many scholars have improved an upper bound for the metric dimension of several graphs or determined their exact values.Chartrand et al. [6] showed the metric dimension of the path graph Susilowati et al. [9] determined the metric dimension of the k-subdivision of the path graph is 1, ≥ 2, cycle graph is 2, ≥ 3, complete graph is n-2, ≥ 4, star graph is n-2, ≥ 3, ladder graph is 2, n ≥ 3 and other special graphs.Borchert et al. [10] computed the metric dimension of the circulant graphs C n (±1, ±2) and demonstrated that if n ≡ 1 (mod 4), then dim(C n (±1,±2)) = 4. Imran et al. [11] investigated the metric dimension of the barycentric subdivision of Möbius ladders, the generalized Petersen multigraphs P(2n, n) and proved that they have metric dimension 3 when n is even and 4 when n is odd.Nawaz et al. [12] demonstrated that the total graph of path power of three Tadpole Graph and Special Trees T( ) and four T( ) has unbounded metric dimension.Nazeer et al. [13] demonstrated that the metric dimension of a two-middle path graph Two−Mid(p q ), q≥3 is 2, threemiddle path graph Three−Mid(P q ), q ≥3 is 3, three-total P q Three−T(P q ), q ≥ 3 is 3, reflection middle tower path graph RL(Tower s ), s ≥3 is 2, middle tower path graph Middle Tower s , s=2 is 1, Middle Tower s , s≥ 3 is 2, symmetrical planar pyramid graph SPPs is 2 and reflection symmetrical planar pyramid graph RL(SPPs) is 2. Ahmad et al. [14] determined the metric dimension of kayak paddles graph KP(ℓ, m, n) and cycles C n with chord and proved that both families possess a metric dimension of 2. Mulyono et al. [15] determined the metric dimension of friendship graph F n , n ≥ 2 is n lollipop graph L m , n , m ≥ 3, n ≥ 1 is m-1 and the Petersen graph P n,m, m = 1, n is odd, n ≥ 3 is 2 and m = 1, n is even, n ≥ 3 is 3. Garces et al. [16] proved that the metric dimension of truncated wheels TW n , n = 3 or n = 6 is 3, n = 4 or n = 5 is 2 and n ≥ 7 is 1.Siddique et al. [17] showed the metric dimensions of antiweb-gear graphs AWJ 2n , n ≥ 15 is and m-level wheel graphs W n,m , n ≥ 7, m ≥ 3 is + 1 .Jäger et al. [18] proved that the metric dimension of Z n × Z n × Z n , n ≥ 2 is .Tomescu et al. [19] determined that the metric dimension of Jahangir graph J 2n , n ≥ 4 is .Tomescu et al. [20] determined that the metric dimension of necklace graph N en is 3 when n is odd and 2 when n is even.The metric dimension of convex polytopes has been studied [21][22][23][24].
In this paper, we determine the metric dimension of Lilly graph and its subdivision, tadpole graph and its subdivision, the subdivisions of the special trees star tree, bistar tree, coconut tree, Y-tree, F-tree and n-centipede tree.
Definition 1.1 [25] The subdivision of a graph is the graph obtained by subdividing each edge of a graph .It is denoted by S ( ).

Figure 1. Line graph G and its subdivision S(G).
We show that dim (G) = 2 and dim(S(G)) = 3.The set B = {v 1 ,v 3 } is a resolving set of G.The representations for the vertices of G with respect to (3,3,3).

Metric Dimension of Lilly Graph L n and Its Subdivision S (Ln)
In this section, we compute the metric dimension of Lilly graph and its subdivision.The metric dimension of Lilly graph and its subdivision have different metric dimension.
is the lower bound.
In conclusion dim (L n ) = .Theorem 2.2 If S(L n ), n ≥ 3 is a subdivision of Lilly graph, then dim (S(L n )) = .

Metric Dimension of Tadpole Graph
In this section, we compute the metric dimension of Tadpole graph and its subdivision.The metric dimension of tadpole graph and its subdivision have the same constant metric dimension 2.  % 1, % 2 , 2 -i -2 ; Case 2. When n is odd.The representations of vertices v i ∈V(T n, m ) in regard to B are As all representations are distinct, dim (T n,m ) = 2. Theorem 3.2 If S(T n,m ), n .6, m .2, is a subdivision of tadpole graph T n,m , then dim(S(T n,m ))=2.Proof.The resolving set in general form is B = {v 1 ,v 2 }⊂ V(S(T n, m )).The representations of vertices v i ∈ V(S(T n, m) ) in regard to B are as follow: Since all vertices have unique representations, we obtain dim (S(T n,m ))= 2.

Metric Dimension of Subdivisions of Special Trees
In this section, we compute the metric dimension of subdivisions of star tree, bistar tree and coconut tree.The metric dimension of star tree and its subdivision have different metric dimension, subdivision of bistar tree S (BT n , n ) has metric dimension  Proof.In [6] it was shown that dim (ST n,n )= n-2 , n .If we choose an ordered set B 1 ⊆B− {v i, v j }, 1-i, jm, i 0j, so that there exist two vertices v i, v j ∈ S (CT (m, n) such that r(v i |B) = r(v j |B) = (3, 3, 3,..., 3, 3).B 1 is not a resolving set, which is contrary to the assumption.As a result, dim S (CT (m, n)) .m is the lower bound.
As a result, dim S (CT (m, n)) = m.

Conclusion
The metric dimension of Lilly graph and its subdivision have different metric dimensions.The tadpole graph and its subdivisions have the same constant metric dimension 2. The metric dimension of star tree and its subdivision have different metric dimension, subdivision of bistar tree S(BT n , n ) has metric dimension *H and subdivision of coconut tree S(CT(m,n)) has metric dimension m.

Theorem 3 . 1
If T n,m , n ≥ 3, m ≥ 1 is a tadpole graph, then dim (T n,m )=2.Proof The tadpole graph T n,m has two cases.

Figure 4 .
Figure 4. Tadpole graph Tn,m.The general form of resolving set of the two cases is B = {v 1 , v 2 }⊂V(T n,m ).Case 1.When n is even.The representations of vertices v i ∈ V(T n, m ) with regard to B are
*H and subdivision of coconut tree S(CT(m,n)) has metric dimension m.Theorem 4.1 If S (ST n,n ), n .6 is a subdivision of star tree then dim (S(ST n,n )) = * .