On Resolvability Parameters of Some Wheel-Related Graphs

Abstract

LetG=V,Ebe a simple connected graph,w∈Vbe a vertex, ande=uv∈Ebe an edge. The distance between the vertexwand edgeeis given byde,w=mindw,u,dw,v, A vertexwdistinguishes two edgese1,e2∈Eifdw,e1≠dw,e2. A setSis said to be resolving if every pair of edges ofGis distinguished by some vertices ofS. A resolving set with minimum cardinality is the basis forG, and this cardinality is the edge metric dimension ofG, denoted byedimG. It has already been proved that the edge metric dimension is an NP-hard problem. The main objective of this article is to study the edge metric dimension of some families of wheel-related graphs and prove that these families have unbounded edge metric dimension. Moreover, the results are compared with the metric dimension of these graphs.