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 dim(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 Ru,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(Ru,v)<1 for any two distinct vertices u,v∈V(G). The fractional metric dimension of G is defined as dimf(G)=min|g|:g is a minimal resolving function of G, where |g|=∑v∈Vg(v). In this paper we study this parameter. © 2011 Elsevier B.V. All rights reserved.
Arumugam, S., & Mathew, V. (2012). The fractional metric dimension of graphs. Discrete Mathematics, 312(9), 1584–1590. https://doi.org/10.1016/j.disc.2011.05.039