PLANARITY AND ITS APPLICATIONS: INVESTIGATING THE BOUNDARIES OF GRAPH EMBEDDINGS

Abstract [English]

Graph theory plays a critical role in numerous applications, particularly in understanding and analyzing the structural properties of networks. This paper focuses on the concept of planarity in graph theory, exploring how certain graphs can be embedded in a plane without edge crossings. The study investigates the criteria that determine graph planarity, including Kuratowski's and Wagner's theorems, and examines the implications of these concepts in various fields such as network design, circuit layout, and geography. Additionally, the paper delves into advanced techniques for testing planarity and embedding graphs in surfaces with higher genus, thus pushing the boundaries of how graph embeddings can be utilized in both theoretical and practical contexts. Through an in-depth analysis of graph embedding algorithms, real-world applications are discussed to highlight the importance of planarity in optimizing spatial layouts and reducing complexity in network structures. The results of this research demonstrate the vast potential of planar graphs in solving complex problems efficiently.