On Fractional Metric Dimension of Graphs

Abstract

A vertex x in a connected graph G = (V, E) is said to resolve a pair {u, v} of vertices of G if the distance from u to x is not equal to the distance from v to x. The resolving neighborhood for the pair {u, v} is defined as R{u, v} = {x ∈ V : d(u, x) ≠ d(v, x)}. A real valued function f : V → [0, 1] is a resolving function (RF) of G if f(R{u, v}) ≥ 1 for any two distinct vertices u, v ∈ V. The weight of f is defined by |f| = f(V) = ∑u∈Vf(v). The fractional metric dimension dim f(G) is defined by dim f(G) = min {|f| : f is a resolving function of G}. In this paper, we characterize graphs G for which [Formula: see text]. We also present several results on fractional metric dimension of Cartesian product of two connected graphs.

The fractional k-metric dimension of graphs

“…An explicit characterization of graphs G satisfying dim f (G) = |V (G)| 2 is given in [3]. We recall the following construction from [3]. Let K = {K n : n ≥ 2} and K = {K n : n ≥ 2}.…”

On fractional metric dimension of comb product graphs

“…By the proof of Lemma 1, if a stem v in G has k ≥ 2 path branches and f is a resolving function of G, then there exists k − 1 path branches of a stem v such that every path branch Z of them satisfies f (V (Z)) ≥ 1 2 . In lemma below, we give an existence of a resolving function f of G such that for every path branch Z of a…”

“…So, g(H(a)) = g(H ′ (a)) + g(P v (a, o)) = g(H ′ (a)). Since an induced subgraph of G £ o H by H ′ (a) is isomorphic to a graph H by deleting all vertices in a path branch P v (o) of v, except the vertex v, according to Lemma 1, we have that g(H ′ (a)) ≥ dim f (H) − 1 2 . Therefore, we obtain g(H(a)) ≥ dim f (H) − 1 2 .…”

“…A formulation of fractional metric dimension as a linear programming problem can be seen in [8]. The fractional metric dimension in graphs was officially initiated by Arumugam and Mathew [1] in 2012. They provided a sufficient condition for a connected graph G whose fractional metric dimension is…”

“…. They also 151 determined dim f (G) where G is Petersen graph, cycles, hypercubes, stars, wheels, friendship graph, and grids [1]. Meanwhile, the fractional metric dimension of trees and unicyclic graphs can be seen in [12].…”