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Abstract
Let 𝐺 = (𝑉, 𝐸) be a connected graph and 𝑆 ⊆ 𝑉(𝐺). For a vertex v ∈ V(G) and an ordered k-partition Π = {𝑆1 , 𝑆2 , … , 𝑆𝑘 } of 𝑉(𝐺), the representation of v with respect to Π is the k-vector 𝑟(𝑣|𝛱 = (𝑑(𝑣, 𝑆1), 𝑑(𝑣, 𝑆2), . . . , 𝑑(𝑣, 𝑆𝑘)), where d(v,Si) denotes the distance between v and Si. The k-partition Π is said to be resolving if for every two vertices 𝑢, 𝑣 𝑉(𝐺), the representation 𝑟(𝑢|П) 𝑟(𝑣|Π). The minimum k for which there is a resolving k-partition of 𝑉(𝐺) is called the partition dimension of 𝐺, denoted by 𝑝𝑑(𝐺). The wheel graph 𝑊𝑛 𝑜𝑛 𝑛 + 1 vertices with 𝑉(𝑊𝑛) = {𝑣0, 𝑣1, . . . , 𝑣𝑛}. Let 𝑙2 ,𝑙2 ,… ,𝑙𝑛be non-negative integers, 𝑙𝑖 ≥ 1, for 𝑖 {0,1,2, . . . , 𝑛}. The thorn graph of the graph Wn, with parameters 𝑙0 ,𝑙1 ,… ,𝑙𝑛 is obtained by attaching li new vertices of degree one to the vertex vi of the graph Wn. The thorn graph is denoted by 𝑇ℎ(𝑊𝑛,𝑙0 ,𝑙1 ,… ,𝑙𝑛). In this paper we give the upper bounds for the partition dimension of 𝑊3 and 𝑊4 denoted by 𝑝𝑑(𝑇ℎ(𝑊3 ,𝑙0 ,𝑙1 ,𝑙2 ,𝑙3 )) and 𝑝𝑑(𝑇ℎ(𝑊4 ,𝑙0 ,𝑙1 ,𝑙2 ,𝑙3 ,𝑙4 )). Keywords : Partition Dimension, Resolving Partition, Thorn Graph, Wheel Graph.