Metric dimension is a graph parameter motivated by problems in robot navigation, drug design, and image processing. In this paper, we answer several open extremal problems on metric dimension and pattern avoidance in graphs from (Geneson, Metric dimension and pattern avoidance, Discrete Appl. Math. 284, 2020, 1-7). Specifically, we construct a new family of graphs that allows us to determine the maximum possible degree of a graph of metric dimension at most $k$, the maximum possible degeneracy of a graph of metric dimension at most $k$, the maximum possible chromatic number of a graph of metric dimension at most $k$, and the maximum $n$ for which there exists a graph of metric dimension at most $k$ that contains $K_{n, n}$. We also investigate a variant of metric dimension called edge metric dimension and solve another problem from the same paper for $n$ sufficiently large by showing that the edge metric dimension of $P_n^{d}$ is $d$ for $n \geq d^{d-1}$. In addition, we use a probabilistic argument to make progress on another open problem from the same paper by showing that the maximum possible clique number of a graph of edge metric dimension at most $k$ is $2^{\Theta(k)}$. We also make progress on a problem from (N. Zubrilina, On the edge dimension of a graph, Discrete Math. 341, 2018, 2083-2088) by finding a family of new triples $(x, y, n)$ for which there exists a graph of metric dimension $x$, edge metric dimension $y$, and order $n$. In particular, we show that for each integer $k > 0$, there exist graphs $G$ with metric dimension $k$, edge metric dimension $3^k(1-o(1))$, and order $3^k(1+o(1))$.

中文翻译：

---

有界度量维度图的极值结果

度量维度是由机器人导航、药物设计和图像处理中的问题驱动的图形参数。在本文中，我们回答了几个关于图中度量维度和模式避免的开放极值问题（Geneson，度量维度和模式避免，离散应用数学。284, 2020, 1-7）。具体来说，我们构建了一个新的图族，它允许我们确定一个度量维度最多为 $k$ 的图的最大可能度数、度量维度最多为 $k$ 的图的最大可能退化程度、最大可能的色度度量维度最多$k$的图的数量，以及存在包含$K_{n, n}$的度量维度最多$k$的图的最大$n$。我们还研究了称为边缘度量维度的度量维度的变体，并通过证明 $P_n^{d}$ 的边缘度量维度对于 $n\geq 是 $d$ 来解决同一篇论文中的另一个问题，因为 $n$ 足够大d^{d-1}$。此外，我们使用概率论在同一篇论文中的另一个开放问题上取得进展，通过证明边度量维度最多为 $k$ 的图的最大可能团数为 $2^{\Theta(k)}$ . 我们还在 (N. Zubrilina, On the edge dimension of a graph, Discrete Math. 341, 2018, 2083-2088) 的一个问题上取得了进展，找到了一个新三元组 $(x, y, n)$ ，其中存在度量维度$x$、边度量维度$y$和阶$n$的图。特别地，我们表明对于每个整数 $k > 0$，存在度量维度为 $k$ 的图 $G$，