The fractional local metric dimension of comb product graphs

Abstract

For the connected graph G with vertex set V(G) and edge set E(G), the local resolving neighborhood Rl{u, v} of two adjacent vertices u, v is defined by Rl{u, v} = {x ∈ V(G): d(x, u) ≠ d(x, v)}. A local resolving function fl of G is a real valued function fl: V(G) → [0,1] such that fl(Rl{u, v}) ≥ 1 for every two adjacent vertices u, v ∈ V(G). The fractional local metric dimension of graph G denoted dimfl(G), is defined by dimfl(G) = min{|ftl|: fl is a local resolving function of G}. One of the operation in graph is the comb product graphs. The comb product graphs of G and H is denoted by G < H. The purpose of this research is to determine the fractional local metric dimension of G < H, for graph G is a connected graph and graph H is a complete graph (Kn). The result of G < Kn is dimfl(G < Kn) = |V(G)|. dimfl(Kn−1).