The partition dimension of cycle books graph

Let G be a nontrivial and connected graph with vertex set V(G), edge set E(G) and S ⊆ V(G) with v ∈ V(G), the distance between v and S is d(v,S) = min{d(v,x)|x ∈ S}. For an ordered partition ∏ = {S₁, S₂, S₃,..., S_k } of V(G), the representation of v with respect to ∏ is defined by r(v|∏) = (d(v, S₁), d(v, S₂),..., d(v, S_k )). The partition ∏ is called a resolving partition of G if all representations of vertices are distinct. The partition dimension pd(G) is the smallest integer k such that G has a resolving partition set with k members. In this research, we will determine the partition dimension of Cycle Books. Cycle books graph is a graph consisting of m copies cycle C_r with the common path P₂. It is shown that the partition dimension of cycle books graph, is 3 for m = 2, 3, and m for m ≥ 4. is 3 + 2k for m = 3k + 2, 4 + 2(k - 1) for m = 3k + 1, and 3 + 2(k - 1) for m = 3k. is m + 1.