Computing the Metric Dimension of Gear Graphs

Let G = (V, E) be a connected graph and d(u, v) denote the distance between the vertices u and v in G. A set of vertices W resolves a graph G if every vertex is uniquely determined by its vector of distances to the vertices in W. A metric dimension of G is the minimum cardinality of a resolving set of G and is denoted by dim(G). Let J2n,m be a m-level gear graph obtained by m-level wheel graph W2n,m ∼= mC2n + k1 by alternatively deleting n spokes of each copy of C2n and J3n be a generalized gear graph obtained by alternately deleting 2n spokes of the wheel graph W3n. In this paper, the metric dimension of certain gear graphs J2n,m and J3n generated by wheel has been computed. Also this study extends the previous result given by Tomescu et al. in 2007.


Introduction and Preliminary Results
In a connected graph G(V, E), where V is the set of vertices and E is the set of edges.The distance d(u, v) between two vertices u, v ∈ V is the length of the shortest path between them and the diameter of G denoted by diam(G) is the maximum distance between pairs of vertices u, v ∈ V(G).Let W = {v 1 , v 2 , . . ., v k } be an order set of vertices of G and u be a vertex of G.The representation r(u|W) of u with respect to W is the k − tuple {d(u, v 1 ), d(u, v 2 ), d(u, v 3 ), . . . ,d(u, v k )}, where W is called a resolving set or locating set if distinct vertices of G have distinct representations with respect to W. See for more results [1,2].
A resolving set of minimum cardinality is called a metric basis for G and the cardinality of a metric basis is said the metric dimension of G, denoted by dim(G), see [3].The motivation for this topic stems from chemistry [4].A common but important problem in the study of chemical structures is to determine ways of representing a set of chemical compounds such that distinct compounds have distinct representations.Moreover the application of this invariant to the navigation of robots in networks are discussed in [5].The application to problems of pattern recognition and image processing, some of which involve the use of hierarchical data structures are given in [6].
For a given ordered set of vertices W = {v 1 , v 2 , . . ., v k } of a graph G, the i th component of r(u|W) is 0 if and only if u = v i .Thus, to show that W is a resolving set it suffices to verify that r(y|W) = r(z|W) for each pair of distinct vertices y, z ∈ V(G)\W.
Motivated by the problem of determining uniquely the location of an intruder in a network, the concept of metric dimension was introduced by Slater in [7] and studied independently by Harary and Melter in [8].
Let Ω be a family of connected graphs F m : Ω = (F m ) m≥1 depending on m as follows: ψ(m) = cardinality of the set of vertices of any member F of Ω and lim m→∞ ψ(m) = ∞.If ∀ m ≥ 1, ∃ C > 0 such that dim(F m ) ≤ C, then we shall say that Ω has bounded metric dimension, otherwise Ω has unbounded metric dimension.If all graphs in Ω have the same metric dimension then F is called a family with constant metric dimension [9].
A connected graph G has dim(G) = 1 if and only if G is a path [5], cycle C n have metric dimension 2 for every n ≥ 3. Other families of graphs with unbounded metric dimension are regular bipartite graphs [10], wheel graph [11].The metric dimensions of m-level wheel graphs, convex polytope graphs and antiweb gear graphs are computed in [12].The metric dimension of honeycomb networks are computed in [13] and t he metric dimension of generators of graphs in [14].In the following section, some results related to m-level generalized gear graph are given.

The Metric Dimension of Double Gear Graph J 2n,m
Definition 1.The joining of two graphs G 1 and G 2 is denoted by G 1 + G 2 with the following vertex and edge sets: Definition 2. In graph theory, an isomorphism of graphs G 1 and G 2 is a bijection between the vertex sets of G 1 and G 2 , f : V(G 1 ) → V(G 2 ) such that any two vertices u and v of G 1 are adjacent in G 1 if and only if f (u) and f (v) are adjacent in G 2 .If an isomorphism exists between two graphs, then the graphs are called isomorphic and denoted as G Note that the the graph C n + K 1 is isomorphic to wheel graph W n .In addition, note that 2C n + K 1 mean union of two copies of C n that are joined with K 1 .Definition 3. A double-wheel graph W n,2 can be obtained as join of 2C n + k 1 and inductively an m-level wheel graph denoted by W n,m can be constructed as W n,m ∼ = mC n + k 1 .Definition 4. A double gear graph denoted by J 2n,2 can be obtained from double-wheel W 2n,2 = 2C 2n + k 1 by alternatively deleting n spokes of each copy of C 2n and inductively an m-level gear graph J 2n,m can be constructed from m-level wheel W 2n,m ∼ = mC 2n + k 1 by alternatively deleting n spokes of each C 2n (see [15]).A double gear graph is depicted in Figure 1.

Construction and Observations
A double gear graph J 2n,2 (see in Figure 1) is constructed if we consider two even cycles with n ≥ 2, C 2n,1 :  The vertices of C 2n,i ; 1 ≤ i ≤ 2, in the graph J 2n,2 are of two kinds namely the vertices of degree 2 and the vertices of degree 3. Vertices of degree 2 and 3 will be considered as minor and major vertices respectively.One can easily check that: with one minor vertex u 2 and two major vertices with three minor vertices u 2 , w 2 , x 2 and one major vertex Consider the gear graph lying between any two neighboring vertices of B are called gaps which are denoted by G i t for 1 ≤ t ≤ r − 1 and G i r , and their cardinalities are said to be the size of gaps.One can easily observe that every vertex of B has two neighboring vertices; gaps generated by these three vertices are called neighboring gaps following a concept already exist in [2] and [17].A gap determined by neighboring vertices of basis say v i and v j will be called an α − β with α ≤ β when deg(v i ) = α and deg(v j ) = β or when deg(v i ) = β and deg(v j ) = α.Hence we have three kinds of gaps namely, 2 − 2 gap, 2 − 3 gap and 3 − 3 gap.
For the graph J 2n,2 , n ≥ 4 central vertex v does not belong to any basis.Since d(v to any metric basis B then there must exists two distinct vertices u i and u j for 1 ≤ i = j ≤ 2n such that r(u i |B) = r(u J |B).Consequently, the basis vertices of J 2n,2 belong to the cycles induced by C 2n,1 and C 2n,2 .It is shown in [17] that if B is a basis of J 2n,1 then B consist only of the vertices of C 2n,1 that satisfy the following properties.

•
If B is a basis of J 2n,1 , n ≥ 6 then every 2 − 2 gap, 2 − 3 gap and 3 − 3 gap of B contains at most 5, 4 and 3 vertices respectively.The vertices of C 2n,i ; 1 ≤ i ≤ 2, in the graph J 2n,2 are of two kinds namely the vertices of degree 2 and the vertices of degree 3. Vertices of degree 2 and 3 will be considered as minor and major vertices respectively.One can easily check that: • When n = 4, dim(J 8,2 ) = 2 + 3, (two minor vertices u 1 , w 1 such that d u 1 , w 1 = 2 of C 2n,1 with one minor vertex u 2 and two major vertices w 2 , x 2 of C 2n,2 such that d u 2 , w 2 = d u 2 , x 2 = 3 form basis).

•
When n = 5, dim(J 10,2 ) = 3 + 4, (three minor vertices u 1 , w 1 , x 1 satisfying d u 1 , with three minor vertices u 2 , w 2 , x 2 and one major vertex Consider the gear graph J 2n,1 in which C 2n,1 is an outer cycle of length 2n.If B is a basis of J 2n,1 then B contains r ≥ 2 vertices of C 2n,1 for n ≥ 6. Suppose B = {v i 1 , v i 2 , . . ., v i r } then vertices of B can be ordered as v i 1 < v i 2 < . . .< v i r such that {v i t , v i t+1 } for 1 ≤ t ≤ r − 1 and {v i r , v i 1 } are called neighboring vertices.Vertices of C 2n,1 lying between any two neighboring vertices of B are called gaps which are denoted by G i t for 1 ≤ t ≤ r − 1 and G i r , and their cardinalities are said to be the size of gaps.One can easily observe that every vertex of B has two neighboring vertices; gaps generated by these three vertices are called neighboring gaps following a concept already exist in [2] and [17].A gap determined by neighboring vertices of basis say v i and v j will be called an α − β with α ≤ β when deg(v i ) = α and deg v j = β or when deg(v i ) = β and deg v j = α.Hence we have three kinds of gaps namely, 2 − 2 gap, 2 − 3 gap and 3 − 3 gap.
For the graph J 2n,2 , n ≥ 4 central vertex v does not belong to any basis.Since d(v j i , v) ≤ 2∀, 1 ≤ i ≤ 2n, 1 ≤ j ≤ 2 and diam(J 2n,2 ) = 4, if central vertex v belongs to any metric basis B then there must exists two distinct vertices u i and u j for 1 ≤ i = j ≤ 2n such that r(u i |B) = r u J B .Consequently, the basis vertices of J 2n,2 belong to the cycles induced by C 2n,1 and C 2n,2 .It is shown in [17] that if B is a basis of J 2n,1 then B consist only of the vertices of C 2n,1 that satisfy the following properties.
• If B is a basis of J 2n,1 , n ≥ 6 then it contains at most one major gap.
• If B is a basis of J 2n,1 , n ≥ 6 then any two neighboring gaps contain together at most six vertices in which one gap is a major gap.

•
If B is a basis of J 2n,1 , n ≥ 6 then any two minor neighboring gaps contain together at most four vertices.Lemma 3. Let B be a basis of J 2n,2 , n ≥ 6, then any two neighboring gaps, one of which being a major gap induced by exactly one of two cycles C 2n,1 or C 2n,2 contain together at most six vertices.

Proof.
If the major gap is 3 − 3 then there is nothing to prove by Lemma 2. Without loss of any generality we can say that only C 2n,1 induced a major gap by Lemma 2. If the major gap is a 2 − 2 gap having five vertices then its neighboring minor gap contains at most one vertex.If this statement is false and 2 − 2 gap, 2 − 3 minor gaps having three and two vertices respectively are neighboring gaps of 2 − 2 major gap, then we have two paths consisting of consecutive vertices of C 2n,1 :u Proof.We have seen that dim(J 8,2 ) = 5 = dim(J 8,1 ) + 8 3 , dim(J 10,2 ) = 7 = dim(J 10,1 ) + 10 3 and the central vertex v does not belong to any basis B of J 2n,2 .Moreover be the outer cycles of J 2n,2 at level 1 and 2 respectively.First we prove that dim(J 2n,2 ) ≤ dim(J 2n,1 ) + 2n 3 by constructing a resolving set W in J 2n,2 with dim(J 2n,1 ) + 2n 3 vertices.We consider three cases according to the residue class modulo 3 to which n belongs.

Case-(ii):
When n ≡ 1(mod3), then we may write 2n = 3k + 2, where k ≥ 4 is even and dim(J 2n,1 ) + 2n 3 = 2k + 1, in this case W can be considered as: Case-(iii): When n ≡ 2(mod3), then we may write 2n = 3k + 1, where k ≥ 5 is even and dim(J 2n,1 ) + 2n 3 = 2k + 1, in this case W can be considered as: The set W contains a unique 2 − 2 major gap having at most five vertices and all other gaps are 2 − 2 minor gaps which contain at most three vertices.The set W is a resolving set of J 2n,2 since any two major or any two minor vertices respectively lying in different gaps or in the same gap are separated by at least one vertex in the set of three vertices of W generating these neighboring gaps.When gaps are not neighboring gaps, then the set of four vertices of W which generate two gaps make the representation unique of each vertex of these two gaps.Representation of central vertex v is (2, 2, 2, . . . , 2),which is different from the representation of all other vertices of J 2n,2 .Hence, Now we show that dim(J 2n,2 ) ≥ dim(J 2n,1 ) + 2n 3 .As the central vertex v does not belong to any basis of J 3n .Let B be a basis of J 2n,2 such that |B|= r then we have r gaps.By lemma 2 B contains at most one major gap, without loss of generality we can say major gap lies on C 2n,1 .Hence B induces r 2 gaps on C 2n,1 and r 2 gaps on C 2n,2 .We denote the gaps on C 2n,1 by G We consider two cases according to the residue class modulo 2 to which r belongs.
1 1 , G 1 2 , G 1 3 , . . ., G 1 r 2 where G 1 i and G 1 i+1 are called neighboring gaps for 1 ≤ i ≤ r 2 − 1 as well as G 1 r 2 is also neighboring gap of G 1 1 and the gaps on C 2n,2 will be denoted by G 2 1 , G 2 2 , G 2 3 , . . ., G 2 r 2 where G 2 i and G 2 i+1 are called neighboring gaps for 1 ≤ i ≤ r 2 − 1 as well as G 2 r 2 is also neighboring gap of G 2 1 .By Lemma 2, suppose G 1 1 is a major gap.By Lemmas 3 and 4, we can write