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On bounded partition dimension of different families of convex polytopes with pendant edges
AIMS Mathematics 2022, 7(3): 4405-4415
Published: 15 March 2021
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Let ψ = ( V , E ) be a simple connected graph. The distance between ρ 1 , ρ 2 V ( ψ ) is the length of a shortest path between ρ 1 and ρ 2 . Let Γ = { Γ 1 , Γ 2 , , Γ j } be an ordered partition of the vertices of ψ. Let ρ 1 V ( ψ ), and r ( ρ 1 | Γ ) = { d ( ρ 1 , Γ 1 ) , d ( ρ 1 , Γ 2 ) , , d ( ρ 1 , Γ j ) } be a j-tuple. If the representation r ( ρ 1 | Γ ) of every ρ 1 V ( ψ ) w.r.t. Γ is unique then Γ is the resolving partition set of vertices of ψ. The minimum value of j in the resolving partition set is known as partition dimension and written as p d ( ψ ) . The problem of computing exact and constant values of partition dimension is hard so one can compute bound for the partition dimension of a general family of graph. In this paper, we studied partition dimension of the some families of convex polytopes with pendant edge such as R n P , D n p and Q n p and proved that these graphs have bounded partition dimension.

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