The fractional k-metric dimension of graphs

Abstract

Let G be a graph with vertex set V (G). For any two distinct vertices x and y of G, let R(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) = Σ s∈U g(s). Let κ(G) = min(|R(x,y)|: x ≠ y and x,y ∈ V (G)). For any real number k ∈ [1; κ(G)], a real-valued function g: V (G) → [0,1] is a k-resolving function of G if g(R(x,y)) ≥ k for any two distinct vertices x,y ∈ V (G). The fractional k-metric dimension, dim kf (G), of G is min(g(V (G)): g is a k-resolving function of G). In this paper, we initi- ate the study of the fractional k-metric dimension of graphs. For a connected graph G and k ∈ [1, κ(G)], it's easy to see that k ≤ dim fk (G) ≤ k|V (G)|/κ(G) ; we characterize graphs G satisfying dim fk (G) = k and dim fk (G) = |V (G)|, respectively. We show that dim fk (G) ≥ k dim f (G) for any k ∈ [1, κ(G)], and we give an example showing that dim fk (G) = k dim f (G) can be arbi- trarily large for some k ∈ (1, κ(G)]; we also describe a condition for which dim fk (G) = k dim f (G) holds. We determine the fractional k-metric dimension for some classes of graphs, and conclude with two open problems, including whether Φ (k) dim fk (G) is a continuous function of k on every connected graph G.