Graph Theory Questions Long
In graph theory, a planar embedding refers to the arrangement of the vertices and edges of a graph in a plane such that no two edges intersect except at their common endpoints. In other words, it is a way of drawing a graph on a plane without any edge crossings.
A planar embedding divides the plane into regions, where each region represents a face of the graph. The outer region, which is unbounded, is called the outer face. The other regions are called inner faces. The number of faces in a planar embedding is denoted by f.
A planar embedding can be represented by a planar graph, which is a graph that can be embedded in a plane without any edge crossings. The planar graph consists of vertices, edges, and faces. The vertices represent the points where the edges intersect, the edges represent the connections between the vertices, and the faces represent the regions bounded by the edges.
The concept of planar embedding is closely related to the planar graph's property of being able to be drawn on a plane without any edge crossings. A graph is said to be planar if it has a planar embedding. The study of planar embeddings and planar graphs is an important area in graph theory, with various applications in computer science, network design, and other fields.
Determining whether a graph has a planar embedding or not can be a challenging problem. However, there are several criteria and algorithms available to test the planarity of a graph, such as Kuratowski's theorem, Euler's formula, and the planarity testing algorithm by Hopcroft and Tarjan.
In summary, a planar embedding is a way of drawing a graph on a plane without any edge crossings, dividing the plane into regions called faces. It is an important concept in graph theory and has various applications in different fields.