The fractional metric dimension of permutation graphs
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2015, Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Chinese Mathematical Society and Springer-Verlag Berlin Heidelberg. 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 RG{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 Ug(s). A real-valued function g: V (G) [0, 1] is a resolving function of G if g(RG{x, y}) 1 for any two distinct vertices x, y V (G). The fractional metric dimension dimf (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 : V (G1) V (G2) be a bijection. Then, a permutation graphG = (V, E) has the vertex set V = V (G1) V (G2) and the edge set E = E(G1) E(G2) {uv | v = (u)}. First, we determine dimf (T) for any tree T. We show that (Formula Presented) 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 > 0, there exists a permutation graph G such that dimf(G) - 1 < . We give examples showing that neither is there a function h1 such that dimf(G) < h1(dimf (G)) for all pairs (G, ), nor is there a function h2 such that h2(dimf (G)) > dimf(G)) for all pairs (G, ). Furthermore, we investigate dimf(G)) when G is a complete k-partite graph or a cycle.
