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 a 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, sdim _f (G), of a graph G is min {g(V(G)): g is a strong resolving function of G}. For any connected graph G of order n ≥ 2, we prove the sharp bounds $1 \le sdim_f(G) \le \frac{n}{2}$. Indeed, we show that sdim _f (G) = 1 if and only if G is a path. If G contains a cut-vertex, then $sdim_f(G) \le \frac{n-1}{2}$ is the sharp bound. We determine sdim _f (G) when G is a tree, a cycle, a wheel, a complete k-partite graph, or the Petersen graph. For any tree T, we prove the sharp inequality sdim _f (T + e) ≥ sdim _f (T) and show that sdim _f (G + e) − sdim _f (G) can be arbitrarily large. Lastly, we furnish a Nordhaus-Gaddum-type result: Let G and $\overline{G}$ (the complement of G) both be connected graphs of order n ≥ 4; it is readily seen that $sdim_f(G)+sdim_f(\overline{G})=2$ if and only if n = 4; further, we characterize unicyclic graphs G attaining $sdim_f(G)+sdim_f(\overline{G})=n$.