The Metric Dimension of Subdivisions of Lilly Graph, Tadpole Graph and Special Trees

IF 4.6 2区数学 Q1 MATHEMATICS, APPLIED

Applied and Computational Mathematics Pub Date : 2023-03-16 DOI:10.11648/j.acm.20231201.12

Basma Mohamed, Mohamed Amin

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引用次数: 2

Abstract

: Consider a robot that is navigating in a space represented by a graph and wants to know its current location. It can send a signal to find out how far it is from each set of fixed landmarks. We study the problem of computing the minimum number of landmarks required, and where they should be placed so that the robot can always determine its location. The set of nodes where the landmarks are located is called the metric basis of the graph, and the number of landmarks is called the metric dimension of the graph. On the other hand, the metric dimension of a graph G is the smallest size of a set B of vertices that can distinguish each vertex pair of G by the shortest-path distance to some vertex in B . The finding of the metric dimension of an arbitrary graph is an NP-complete problem. Also, the metric dimension has several applications in different areas, such as geographical routing protocols, network discovery and verification, pattern recognition, image processing, and combinatorial optimization. In this paper, we study the metric dimension of subdivisions of several graphs, including the Lilly graph, the Tadpole graph, and the special trees star tree, bistar tree, and coconut tree.

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Lilly图、蝌蚪图和特殊树的细分度量维数

:考虑一个机器人在一个图形表示的空间中导航，并想知道它的当前位置。它可以发送一个信号来确定它离每组固定地标有多远。我们研究了计算所需的最小地标数量的问题，以及它们应该放置在哪里，以便机器人始终能够确定其位置。地标所在节点的集合称为图的度量基，地标的个数称为图的度量维数。另一方面，图G的度量维数是一个顶点集合B的最小尺寸，它可以通过到B中某个顶点的最短路径距离来区分G的每个顶点对。求任意图的度量维数是一个np完全问题。此外，度量维度在不同的领域有几个应用，例如地理路由协议、网络发现和验证、模式识别、图像处理和组合优化。本文研究了几种图的细分度量维数，包括Lilly图、Tadpole图以及特殊树——星树、双星树和椰子树。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Applied and Computational Mathematics 数学-应用数学

CiteScore

8.80

自引率

5.00%

发文量

审稿时长

6 months

期刊介绍： Applied and Computational Mathematics (ISSN Online: 2328-5613, ISSN Print: 2328-5605) is a prestigious journal that focuses on the field of applied and computational mathematics. It is driven by the computational revolution and places a strong emphasis on innovative applied mathematics with potential for real-world applicability and practicality. The journal caters to a broad audience of applied mathematicians and scientists who are interested in the advancement of mathematical principles and practical aspects of computational mathematics. Researchers from various disciplines can benefit from the diverse range of topics covered in ACM. To ensure the publication of high-quality content, all research articles undergo a rigorous peer review process. This process includes an initial screening by the editors and anonymous evaluation by expert reviewers. This guarantees that only the most valuable and accurate research is published in ACM.