For an ordered subset W = {w 1 , w 2 , w 3 ,. .. , w k } of vertices in a connected graph G and a vertex v of G, the metric representation of v with respect to W is the k-vector r(v|W) = (d(v, w 1), d(v, w 2), d(v, w 3),. .. , d(v, w k)). The set W is called a resolving set of G if r(u|W) = r(v|W) implies u = v. The metric dimension of G, denoted by β(G), is the minimum cardinality of a resolving set of G. Let n ≥ 3 be an integer. A truncated wheel, denoted by T W n , is the graph with vertex set V (T W n) = {a} ∪ B ∪ C, where B = {b i : 1 ≤ i ≤ n} and C = {c j,k : 1 ≤ j ≤ n, 1 ≤ k ≤ 2}, and edge set E(T W n) = {ab i : 1 ≤ i ≤ n} ∪ {b i c i,k : 1 ≤ i ≤ n, 1 ≤ k ≤ 2} ∪ {c j,1 c j,2 : 1 ≤ j ≤ n} ∪ {c j,2 c j+1,1 : 1 ≤ j ≤ n}, where c n+1,1 = c 1,1. In this paper, we compute the metric dimension of truncated wheels.

Abstract. For an ordered subset W = {w1, w2, . . . , w k } of vertices in a connected graph G and a vertex v of G, the metric representation of v with respect to W is the k-vectoris the distance of the vertices v and wi in G. The set W is called a resolving set of G if r(u|W ) = r(v|W ) implies u = v. The metric dimension of G, denoted by β(G), is the minimum cardinality of a resolving set of G, and a resolving set of G with cardinality equal to its metric dimension is called a metric basis of G. A set T of vectors is called a positive lattice set if all the coordinates in each vector of T are positive integers. A positive lattice set T consisting of n k-vectors is called a metric graphic set if there exists a simple connected graph G of order n + k with β(G) = k such that T = {r(u i|S) : ui ∈ V (G)\S, 1 ≤ i ≤ n} for some metric basis S = {s1, s2, . . . , s k } of G. If such G exists, then we say G is a metric graphic realization of T . In this paper, we introduce the concept of metric graphic sets anchored on the concept of metric dimension and provide some characterizations. We also give necessary and sufficient conditions for any positive lattice set consisting of 2 k-vectors to be a metric graphic set. We provide an upper bound for the sum of all the coordinates of any metric graphic set and enumerate some properties of positive lattice sets consisting of n 2-vectors that are not metric graphic sets.

Let G be a connected graph. For a vertex v of G and a subset S of V(G), the distance between v and S is d(v, S) ¼ minfdðv, xÞ, x 2 Sg: Given an ordered k-partition P¼fS 1 , S 2 , :::, S k g of V(G), the representation of v with respect to P is the k-vector rðvjPÞ ¼ ðdðv, S 1 Þ, dðv, S 2 Þ, :::, dðv, S k ÞÞ: If rðujPÞ 6 ¼ rðvjPÞ for each pair of distinct vertices u, v 2 VðGÞ, then the k-partition P is said to be a resolving partition. The partition dimension of G, denoted by pd(G), is determined by the minimum k for which there is a resolving partition of V(G). If each induced subgraph hS i i for S i , 1 i k, is connected in G, then the resolving partition P¼fS 1 , S 2 , :::, S k g of V(G) is said to be connected. The connected partition dimension of G, denoted by cpd(G), is determined by the minimum k for which there is a connected resolving partition of V(G). In this paper, we compute the connected partition dimension of the truncated wheels TW n . It is shown that for any natural number n ! 3, the connected partition dimension of the truncated wheel TW n is 3 when n ¼ 3 and dn=3e þ 1 when n ! 4:

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