The fractional metric dimension of graphs

The fractional k-metric dimension of graphs

“…Next, we recall some results on the fractional metric dimension of graphs. One can easily see that, for any connected graph G of order at least two, 1 ≤ dim f (G) ≤ |V (G)| 2 (see [2]). For the characterization of graphs G achieving the lower bound, see Theorem 2.7(a).…”

“…For the characterization of graphs G achieving the lower bound, see Theorem 2.7(a). Regarding the characterization of graphs G achieving the upper bound, the following result is stated in [2] and a correct proof is provided in [21].…”

“…A formulation of fractional metric dimension as a linear programming problem can be found in [15]. Arumugam and Mathew [2] officially studied the fractional metric dimension of graphs. For a function g defined on V (G) and for U ⊆ V (G), let…”

“…[2,21] Let G be a connected graph of order at least two. Then dim f (G) = |V (G)| 2 if and only if there exists a bijection α :…”

“…Note that κ(W 5 ) = 2 since R{u 1 , u 3 } = {u 1 , u 3 }. Let k ∈[1,2]. Let g : V (W 5 ) → [0, 1] be a function defined by g(v) = 0 and g(u i ) = 1 2 for each i ∈ {1, 2, 3, 4}.…”