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arXiv 2401.00705math.CO
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关于树的立方图的度量维数

On metric dimension of cube of trees

Sanchita Paul, Bapan Das, Avishek Adhikari, Laxman Saha

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AI总结：

本文研究了树的立方图 $T^3$ 的度量维数，建立了顶点子集成为分辨集的充要条件以确定其紧致上下界，并刻画了毛毛虫图、龙虾图等特定树立方图的度量维数边界及取值构造。

摘要：

设 $G=(V,E)$ 为一个连通图，$d_{G}(u,v)$ 为 $G$ 中顶点 $u$ 和 $v$ 之间的最短距离。集合 $S=\{s_{1},s_{2},\cdots,s_{n}\}\subset V(G)$ 被称为一个 resolving set（分辨集），如果对于 $G$ 的所有不同顶点 $u,v$，存在一个元素 $s\in S$ 使得 $d(s,u)\neq d(s,v)$。图 $G$ 的 resolving set 的最小基数被称为 $G$ 的 metric dimension（度量维数），记为 $β{(G)}$。具有 $β{(G)}$ 个顶点的 resolving set 被命名为 $G$ 的 metric basis（度量基）。度量维数问题是在图 $G$ 中寻找一个 metric basis，它在网络理论、电信、图像处理、模式识别等许多其他领域有若干现实应用。在本文中，我们考虑 cube of trees（树的立方图） $T^{3}=(V, E)$，其中任意两个顶点 $u,v$ 相邻当且仅当它们在 $T$ 中的距离小于等于三。我们建立了 $V$ 的顶点子集成为 $T^{3}$ 的 resolving set 的充分必要条件。这有助于确定 $T^{3}$ 的度量维数的紧致界限（上界和下界）。然后，对于某些著名的树的立方图，例如 caterpillars（毛毛虫图）、lobsters（龙虾图）、spiders（蜘蛛图）和 $d$-regular trees（$d$-正则树），我们确立了度量维数的边界。此外，我们刻画了满足 $β{(T^{3})}=β{(T)}$ 的树的立方图的一些受限族。我们提供了一个构造，展示了存在一个树的立方图，其度量维数可以取到每一个正整数值。

英文摘要：

Let $G=(V,E)$ be a connected graph and $d_{G}(u,v)$ be the shortest distance between the vertices $u$ and $v$ in $G$. A set $S=\{s_{1},s_{2},\cdots,s_{n}\}\subset V(G)$ is said to be a {\em resolving set} if for all distinct vertices $u,v$ of $G$, there exist an element $s\in S$ such that $d(s,u)\neq d(s,v)$. The minimum cardinality of a resolving set for a graph $G$ is called the {\em metric dimension} of $G$ and it is denoted by $β{(G)}$. A resolving set having $β{(G)}$ number of vertices is named as {\em metric basis} of $G$. The metric dimension problem is to find a metric basis in a graph $G$, and it has several real-life applications in network theory, telecommunication, image processing, pattern recognition, and many other fields. In this article, we consider {\em cube of trees} $T^{3}=(V, E)$, where any two vertices $u,v$ are adjacent if and only if the distance between them is less than equal to three in $T$. We establish the necessary and sufficient conditions of a vertex subset of $V$ to become a resolving set for $T^{3}$. This helps determine the tight bounds (upper and lower) for the metric dimension of $T^{3}$. Then, for certain well-known cubes of trees, such as caterpillars, lobsters, spiders, and $d$-regular trees, we establish the boundaries of the metric dimension. Further, we characterize some restricted families of cube of trees satisfying $β{(T^{3})}=β{(T)}$. We provide a construction showing the existence of a cube of tree attaining every positive integer value as their metric dimension.

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