On the fractional strong metric dimension of graphs

Abstract

For any two vertices x and y of a graph G, let S{x,y} denote the set of vertices z such that either x lies on a y−z geodesic or y lies on an x−z geodesic. For a function g defined on V(G) and U⊆V(G), let g(U)=∑x∈Ug(x). A function g:V(G)→[0,1] is a strong resolving function of G if g(S{x,y})≥1, for every pair of distinct vertices x,y of G. The fractional strong metric dimension, sdimf(G), of a graph G is min{g(V(G)):g is a strong resolving function of G}. This paper furthers the study of fractional strong metric dimension initiated in COCOA 2013 (Lecture Notes in Comput. Sci.). First, we clarify or correct the proofs to two characterization theorems contained in two papers on fractional (strong) metric dimension. Next, results on fractional strong metric dimension analogous to the work of Feng, Lv, and Wang on fractional metric dimension are offered. We provide new upper and lower bounds on sdimf(G), partly in analogy with the work done by Feng et al. and partly by exploiting the particular nature of the strong metric dimension. Finally, motivated by the work of Arumugam, Mathew, and Shen, we describe a class of graphs G for which sdimf(G)=|V(G)|2.