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.

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Let G = (V, E) be a connected graph and let k be a positive integer with k ≤ rad(G). A subset D ⊆ V is called a distance k-dominating set of G if for every v ∈ V − D, there exists a vertex u ∈ D such that d(u, v) ≤ k. In this paper we study the fractional version of distance k-domination and related parameters.

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