The fractional metric dimension of graphs

A vertex x in a connected graph G 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 . A set S of vertices of G is a resolving set for G if every pair of vertices is resolved by some vertex of S . The smallest cardinality of a resolving set for G , denoted by d i m ( G ) , is called the metric dimension of G . For the pair { u , v } of vertices of G the collection of all vertices which resolve the pair { u , v } is denoted by R { u , v } and is called the resolving neighbourhood of the pair { u , v } . A real valued function g : V ( G ) → [ 0 , 1 ] is a resolving function of G if g ( R { u , v } ) ≥ 1 for any two distinct vertices u , v ∈ V ( G ) . The fractional metric dimension of G is defined as d i m f ( G ) = min { | g | : g  is a minimal resolving function of  G } , where | g | = ∑ v ∈ V g ( v ) . In this paper we study this parameter.