图的分数$k$-截断度量维数
The fractional $k$-truncated metric dimension of graphs
摘要：
图$G$的度量维数$\dim(G)$和分数度量维数$\dim_f(G)$已被广泛研究。设$G$是一个顶点集为$V(G)$的图，$d(x,y)$表示$G$中最短$x-y$路的长度。令$k$为正整数。对于任何$x,y \in V(G)$，设为$d_k(x,y)=\min\{d(x,y), k+1\}$，设为$R_k\{x,y\}=\{z\in V(G): d_k(x,z) \neq d_k(y,z)\}$。一个集合$S \subseteq V(G)$是$G$的一个\emph{$k$-truncated resolving set}，如果对于任何不同的$x,y\in V(G)$有$|S \cap R_k\{x,y\}| \ge 1$，并且$G$的\emph{$k$-truncated metric dimension} $\dim_k(G)$是$G$的所有$k$-截断解析集上的最小基数。对于在$V(G)$上定义的函数$g$和$U \subseteq V(G)$，设$g(U)=\sum_{s\in U}g(s)$。实值函数$g:V(G) \rightarrow[0,1]$是$G$的\emph{$k$-truncated resolving function}，如果$g(R_k\{x,y\}) \ge 1$对于任何不同的$x, y\in V(G)$，并且$G$的\emph{fractional $k$-truncated metric dimension} $\dim_{k,f}(G)$是$\min\{g(V(G)): g \mbox{ is a $ k $-truncated resolving function of }G\}$。请注意，如果$k$截断的解析函数的余域被限制为$\{0,1\}$，则$\dim_{k,f}(G)$简化为$\dim_k(G)$，如果$k$至少是$G$的直径，则$\dim_{k,f}(G)=\dim_f(G)$。本文研究了图的分数$k$-截断度量维数.对任意阶为$n\ge2$的连通图$G$，我们证明了$1 \le \dim_{k,f}(G) \le \frac{n}{2}$;我们刻画了满足$\dim_{k,f}(G)$的$G$分别等于$1$和$\frac{n}{2}$.我们研究$\dim_{k,f}(G)$的一些图类。我们还证明了非同构图$G$和$H$的存在性，使得$\dim_k(G)=\dim_k(H)$和$\dim_{k,f}(G)\neq \dim_{k,f}(H)$，并研究了$\dim(G)$，$\dim_f(G)$，$\dim_k(G)$和$\dim_{k,f}(G)$之间的关系.我们总结了一些开放的问题的文件。
英文摘要：
The metric dimension, $\dim(G)$, and the fractional metric dimension, $\dim_f(G)$, of a graph $G$ have been studied extensively. 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 $R_k\{x,y\}=\{z\in V(G): d_k(x,z) \neq d_k(y,z)\}$. A set $S \subseteq V(G)$ is a \emph{$k$-truncated resolving set} of $G$ if $|S \cap R_k\{x,y\}| \ge 1$ for any distinct $x,y\in V(G)$, and the \emph{$k$-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 \emph{$k$-truncated resolving function} of $G$ if $g(R_k\{x,y\}) \ge 1$ for any distinct $x, y\in V(G)$, and the \emph{fractional $k$-truncated metric dimension} $\dim_{k,f}(G)$ of $G$ is $\min\{g(V(G)): g \mbox{ is a $k$-truncated resolving function of }G\}$. Note that $\dim_{k,f}(G)$ reduces to $\dim_k(G)$ if the codomain of $k$-truncated resolving functions is restricted to $\{0,1\}$, and $\dim_{k,f}(G)=\dim_f(G)$ if $k$ is at least the diameter of $G$. In this paper, we study the fractional $k$-truncated metric dimension of graphs. For any connected graph $G$ of order $n\ge2$, 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 examine $\dim_{k,f}(G)$ of some graph classes. We also show the existence of non-isomorphic graphs $G$ and $H$ such that $\dim_k(G)=\dim_k(H)$ and $\dim_{k,f}(G)\neq \dim_{k,f}(H)$, and we examine the relation among $\dim(G)$, $\dim_f(G)$, $\dim_k(G)$ and $\dim_{k,f}(G)$. We conclude the paper with some open problems.
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