Bounds on partition dimension of Peterson graphs

Abstract

The distance of a connected, simple graph ℙ is denoted by d(η 1, η 2), which is the length of a shortest path between the vertices η 1, η 2 ∈ V(ℙ), where V(ℙ) is the vertex set of ℙ. The l - ordered partition of V(ℙ) is θ = {θ 1, θ 2, … , θ l }. A vertex η ∈ V(ℙ), and r(η|θ) = {d(η,θ 1), d(η,θ 2), … , d(η,θ l )} be a l - tuple distances, where r(η|θ) is the representation of a vertex η with respect to set θ. If r(η|θ) of η is unique, for every pair of vertices, then θ is the resolving partition set of V(ℙ). The minimum number l in the resolving partition set θ is known as partition dimension (pd(ℙ)). In this paper, we studied the generalized families of Peterson graph, P λ,λ-1 and proved that these families have bounded partition dimension.