caa a22 4500 605461589 CHVBK 20210128100244.0 cr unu---uuuuu 210128e20150301xx s 000 0 eng 10.1007/s10114-015-4160-5 doi (NATIONALLICENCE)springer-10.1007/s10114-015-4160-5 Yi Eunjeong Texas A&M University at Galveston, 77553, Galveston, TX, USA aut The fractional metric dimension of permutation graphs [Elektronische Daten] [Eunjeong Yi] 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) = Σ 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 G 1 and G 2 be disjoint copies of a graph G, and let σ: V (G 1) → V (G 2) be a bijection. Then, a permutation graph G σ = (V, E) has the vertex set V = V (G 1) ∪ V (G 2) and the edge set E = E(G 1) ∪ E(G 2) ∪ {uv | v = σ(u)}. First, we determine dimf (T) for any tree T. We show that $1 < \dim _f (G_\sigma ) \leqslant \tfrac{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 ɛ > 0, there exists a permutation graph G σ such that dim f (G σ) - 1 < ε. We give examples showing that neither is there a function h 1 such that dim f (G) < h 1(dim f (G σ)) for all pairs (G, σ), nor is there a function h 2 such that h 2(dim f (G)) > dim f (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. Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Chinese Mathematical Society and Springer-Verlag Berlin Heidelberg, 2015 Fractional metric dimension nationallicence permutation graph nationallicence tree nationallicence complete k -partite graph nationallicence cycle nationallicence Acta Mathematica Sinica, English Series Institute of Mathematics, Chinese Academy of Sciences and Chinese Mathematical Society 31/3(2015-03-01), 367-382 1439-8516 31:3<367 2015 31 10114 https://doi.org/10.1007/s10114-015-4160-5 text/html Onlinezugriff via DOI BK010053 XK010053 XK010000 Metadata rights reserved Springer special CC-BY-NC licence nationallicence 1 research-article jats NATIONALLICENCE NATIONALLICENCE NL-springer NATIONALLICENCE 856 40 https://doi.org/10.1007/s10114-015-4160-5 text/html Onlinezugriff via DOI NATIONALLICENCE 100 1- Yi Eunjeong Texas A&M University at Galveston, 77553, Galveston, TX, USA aut NATIONALLICENCE 773 0- Acta Mathematica Sinica, English Series Institute of Mathematics, Chinese Academy of Sciences and Chinese Mathematical Society 31/3(2015-03-01), 367-382 1439-8516 31:3<367 2015 31 10114