The fractional k-metric dimension of graphs
Cong X. Kang, Ismael G. Yero, Eunjeong Yi · Applicable Analysis and Discrete Mathematics · 2018
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)= ? s?U g(s). Let k(G) = min{|R{x,y}|: x ? y and x,y ? V (G)}. For any real number k ?[1,k(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, dimkf(G), of G is min{g(V(G)):g is a k-resolving function of G}. In this paper, we initiate the study of the fractional k-metric dimension of graphs. For a connected graph G and k?[1,k(G)], it's easy to see that k ? dimkf(G)? k|V(G)|/k(G); we characterize graphs G satisfying dimkf(G)=k and dimkf(G)=|V(G)| respectively. We show that dimkf(G) ? k dimf(G) for any k ? [1,k(G)], and we give an example showing that dimkf(G)- k dimf(G) can be arbitrarily large for some k?(1,k(G)]; we also describe a condition for which dimkf(G) = kdimf(G) holds. We determine the fractional k-metric dimension for some classes of graphs, and conclude with two open problems, including whether ?(k) = dimkf(G) is a continuous function of k on every connected graph G.