Bounds on partition dimension of Peterson graphs. Issue 7 (3rd October 2021) - Khalaf, Abdul Jalil M.; Nadeem, Muhammad Faisal; Azeem, Muhammasd; Cancan, Murat; Farahani, Mohammad Reza
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.