Muhammad Faisal Nadeem, Ayesha Shabbir, Muhammad Imran
{"title":"Optimal Embedding of Graphs with Nonconcurrent Longest Paths in Archimedean Tessellations","authors":"Muhammad Faisal Nadeem, Ayesha Shabbir, Muhammad Imran","doi":"10.1155/2023/3919389","DOIUrl":null,"url":null,"abstract":"<div>\n <p>Optimal graph embeddings represent graphs in a lower dimensional space in a way that preserves the structure and properties of the original graph. These techniques have wide applications in fields such as machine learning, data mining, and network analysis. Do we have small (if possible minimal) <i>k</i>-connected graphs with the property that for any <i>j</i> vertices there is a longest path avoiding all of them? This question of Zamfirescu (1972) was the first variant of Gallai’s question (1966): Do all longest paths in a connected graph share a common vertex? Several good examples answering Zamfirescu’s question are known. In 2001, he asked to investigate the family of geometrical lattices with respect to this property. In 2017, Chang and Yuan proved the existence of such graphs in Archimedean tiling. Here, we prove that the graphs presented by Chang and Yuan are not optimal by constructing such graphs of sufficiently smaller orders. The problem of finding nonconcurrent longest paths in Archimedean tessellations refers to finding paths in a lattice such that the paths do not overlap or intersect with each other and are as long as possible. The complexity of embedding graph is still unknown. This problem can be challenging because it requires finding paths that are both long and do not intersect, which can be difficult due to the constraints of the lattice structure.</p>\n </div>","PeriodicalId":50653,"journal":{"name":"Complexity","volume":"2023 1","pages":""},"PeriodicalIF":1.7000,"publicationDate":"2023-07-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1155/2023/3919389","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Complexity","FirstCategoryId":"5","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1155/2023/3919389","RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS, INTERDISCIPLINARY APPLICATIONS","Score":null,"Total":0}
引用次数: 0
Abstract
Optimal graph embeddings represent graphs in a lower dimensional space in a way that preserves the structure and properties of the original graph. These techniques have wide applications in fields such as machine learning, data mining, and network analysis. Do we have small (if possible minimal) k-connected graphs with the property that for any j vertices there is a longest path avoiding all of them? This question of Zamfirescu (1972) was the first variant of Gallai’s question (1966): Do all longest paths in a connected graph share a common vertex? Several good examples answering Zamfirescu’s question are known. In 2001, he asked to investigate the family of geometrical lattices with respect to this property. In 2017, Chang and Yuan proved the existence of such graphs in Archimedean tiling. Here, we prove that the graphs presented by Chang and Yuan are not optimal by constructing such graphs of sufficiently smaller orders. The problem of finding nonconcurrent longest paths in Archimedean tessellations refers to finding paths in a lattice such that the paths do not overlap or intersect with each other and are as long as possible. The complexity of embedding graph is still unknown. This problem can be challenging because it requires finding paths that are both long and do not intersect, which can be difficult due to the constraints of the lattice structure.
期刊介绍:
Complexity is a cross-disciplinary journal focusing on the rapidly expanding science of complex adaptive systems. The purpose of the journal is to advance the science of complexity. Articles may deal with such methodological themes as chaos, genetic algorithms, cellular automata, neural networks, and evolutionary game theory. Papers treating applications in any area of natural science or human endeavor are welcome, and especially encouraged are papers integrating conceptual themes and applications that cross traditional disciplinary boundaries. Complexity is not meant to serve as a forum for speculation and vague analogies between words like “chaos,” “self-organization,” and “emergence” that are often used in completely different ways in science and in daily life.