论文标题
边缘度量维度小于度量尺寸的图形
Graphs with the edge metric dimension smaller than the metric dimension
论文作者
论文摘要
鉴于连接的图形$ g $，$ g $的度量标准（分别为edge度量）尺寸是最小有序的一组顶点的基数，它们通过距离向量唯一地识别$ g $的每对不同的顶点（分别边缘）的基础。在这项工作中，我们在图形的（边缘）度量维度上解决了三个开放问题。具体来说，我们表明，对于每$ r，t \ ge 2 $，带有$ r \ ne t $，都有$ n_0 $，以至于每$ n \ ge n_0 $都存在一个图$ g $ agragh $ g $ agrage $ n $ a $ n $ n $ n $ n $ n $ n $ n $ n $ r $ r $ r $ r $ and Edge $ t $ t $，在其他后果中显示了它的数量较小的尺寸，它的数量较小。此外，我们还证明，不可能将图$ g $的边缘度量尺寸限制为$ g $的度量尺寸的一定恒定因素。
Given a connected graph $G$, the metric (resp. edge metric) dimension of $G$ is the cardinality of the smallest ordered set of vertices that uniquely identifies every pair of distinct vertices (resp. edges) of $G$ by means of distance vectors to such a set. In this work, we settle three open problems on (edge) metric dimension of graphs. Specifically, we show that for every $r,t\ge 2$ with $r\ne t$, there is $n_0$, such that for every $n\ge n_0$ there exists a graph $G$ of order $n$ with metric dimension $r$ and edge metric dimension $t$, which among other consequences, shows the existence of infinitely many graph whose edge metric dimension is strictly smaller than its metric dimension. In addition, we also prove that it is not possible to bound the edge metric dimension of a graph $G$ by some constant factor of the metric dimension of $G$.
