The Fractional k-truncated Metric Dimension of Graphs

Abstract

Let G be a graph with vertex set V(G), and let d(x, y) denote the length of a shortest $x-y$ path in G. Let k be a positive integer. For any $x,y \in V(G)$, let $d_k(x,y)=\min \{d(x,y), k+1\}$ and let $R_k\{x,y\}=\{z\in V(G): d_k(x,z) \ne d_k(y,z)\}$. A set $S \subseteq V(G)$ is a k-truncated resolving set of G if $|S \cap R_k\{x,y\}| \ge 1$ for any distinct $x,y\in V(G)$, and the k-truncated metric dimension $\dim _k(G)$ of G is the minimum cardinality over all k-truncated resolving sets of G. For a function g defined on V(G) and for $U \subseteq V(G)$, let $g(U)=\sum _{s\in U}g(s)$. A real-valued function $g:V(G) \rightarrow [0,1]$ is a k-truncated resolving function of G if $g(R_k\{x,y\}) \ge 1$ for any distinct $x, y\in V(G)$, and the fractional k-truncated metric dimension $\dim _{k,f}(G)$ of G is $\min \{g(V(G)): g \text{ is } \text{ a } k\text{-truncated } \text{ resolving } \text{ function } \text{ of } G\}$. Note that $\dim _{k,f}(G)$ reduces to $\dim _k(G)$ if the codomain of k-truncated resolving functions is restricted to $\{0,1\}$. In this paper, we initiate the study of the fractional k-truncated metric dimension of graphs. For any connected graph G of order $n\ge 2$, we show that $1 \le \dim _{k,f}(G) \le \frac{n}{2}$; we characterize G satisfying $\dim _{k,f}(G)$ equals 1 and $\frac{n}{2}$, respectively. We also examine $\dim _{k,f}(G)$ of some graph classes and conclude with some open problems.