DefinePK

Abstract

For a simple connected graph G, the distance d(ai, bi), where ai, bi ∈ V (G), is the length of the shortest path, measured by the number of edges, between vertices ai and bi. The n-order partition of vertices of G is denoted as ψ = {ψ1, ψ2, ψ3, . . . , ψn}. Thenotation of vertex ai ∈ G with respect to ψ is the vector code {d(ai, ψ1), d(ai, ψ2), . . . , d(ai, ψn)}.Partition set ψ is called a resolving partition set if the representation of each vertex with respect to ψ is unique. The partition dimension of G is defined as the minimum size of such a resolving partition set. In this research, we investigated the partition di- mension of the generalized gear graph G(k, n) and the generalized fan graph F (2, n)..