The fractional metric dimension of permutation graphs

Let G= (V (G),E(G)) be a graph with vertex set V (G) and edge set E(G). For two distinct vertices x and y of a graph G, let R_G{x, y} denote the set of vertices z such that the distance from x to z is not equal to the distance from y to z in G. For a function g defined on V (G) and for U ⊆ V (G), let g(U) =∑ _s∈U g(s). A real-valued function g: V (G) → 0, 1] is a resolving function of G if g(R_G{x, y}) ≥ 1 for any two distinct vertices x, y ∈ V (G). The fractional metric dimension dim_f (G) of a graph G is min{g(V (G)): g is a resolving function of G}. Let G₁ and G₂ be disjoint copies of a graph G, and let σ: V (G₁) → V (G₂) be a bijection. Then, a permutation graph G_σ = (V,E) has the vertex set V = V (G₁) ∪ V (G₂) and the edge set E = E(G₁) ∪ E(G₂) ∪ {uv | v = σ(u)}. First, we determine dim_f (T) for any tree T. We show that 1 < dim_f (G_σ) ≤ 1/ 2 (|V (G)| + |S(G)|) for any connected graph G of order at least 3, where S(G) denotes the set of support vertices of G. We also show that, for any ε > 0, there exists a permutation graph G_σ such that dimf (G_σ) - 1 < ε. We give examples showing that neither is there a function h₁ such that dim_f (G) < h₁ (_f (G_σ)) for all pairs (G, σ), nor is there a function h₂ such that h₂(dim_f (G)) > dimf (G_σ) for all pairs (G, σ). Furthermore, we investigate dim_f (G_σ) when G is a complete k-partite graph or a cycle.