Let be a simple connected graph with a set of vertices ( ) and set of edges ( ). The distance between two vertices and in a graph are the shortest path length between two vertices and denoted by ( , ). Let be a positive integer, ⊆ with is a -metric generator if and only if for each different vertex pair , ∈ there are at least vertices 1 , 2 , … , ∈ and fulfill ( , ) ≠ ( , ) with ∈ {1, 2, … , }. Minimum cardinality of a -metric generator of a graph is called the basis -metric of graph . The number of elements on the basis of -metric graph are called -metric dimension of graph and denoted by ( ). + is the result of a join operation between null graph and path graph with , ≥ 2. Starbarbell graph denoted by 1, 2,…, is a graph formed from a star graph 1, and complete graph then merge one vertex from each with ℎ leaf of 1, with ≥ 3, 1 ≤ ≤ , and ≥ 2. In this paper, we determine the -metric dimension of + graph and starbarbell graph.
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