Planar Graph

Definition

A planar graph is a graph that can be drawn on a plane without any edges crossing each other. This concept is crucial as it connects various mathematical principles, like Euler's formula, which relates the number of vertices, edges, and faces in a connected planar graph. Understanding planar graphs also involves techniques for testing their planarity and finding embeddings, as well as applications in polygon triangulation for efficient computational geometry.

5 Must Know Facts For Your Next Test

1. Euler's formula for connected planar graphs states that $$V - E + F = 2$$, where $$V$$ is the number of vertices, $$E$$ is the number of edges, and $$F$$ is the number of faces.
2. A graph can have multiple planar embeddings, which means there can be different ways to draw it on a plane without crossings.
3. Planarity testing algorithms exist to determine whether a graph can be drawn without edge crossings, which are essential for applications in network design.
4. Triangulation of polygons involves dividing the polygon into triangles using non-crossing diagonals, and this process often relies on the properties of planar graphs.
5. Kuratowski's theorem provides a criterion for determining whether a graph is planar by identifying forbidden subgraphs.

Review Questions

• How does Euler's formula apply to planar graphs and what implications does it have for understanding their structure?
  • Euler's formula states that for any connected planar graph, the relationship between the number of vertices (V), edges (E), and faces (F) can be expressed as $$V - E + F = 2$$. This means that if you know any two of these quantities, you can find the third. It helps us understand how changes in one aspect, like adding edges or vertices, will impact the overall structure of the graph. This formula is fundamental in studying properties like connectivity and surface representation.
• Discuss the role of Kuratowski's theorem in determining if a graph is planar.
  • Kuratowski's theorem plays a key role in characterizing planar graphs by stating that a graph is non-planar if it contains a subgraph that is a subdivision of K5 (the complete graph on five vertices) or K3,3 (the complete bipartite graph with three vertices in each partition). This theorem provides a practical approach to identifying non-planarity by searching for these specific configurations within any given graph. It highlights the connection between certain structures and planarity, making it easier to analyze complex graphs.
• Evaluate how triangulating polygons relates to the concept of planar graphs and its applications in computational geometry.
  • Triangulating polygons involves dividing them into triangles using diagonals without crossings, which directly ties into the study of planar graphs. Each triangle formed can be represented as a face in a planar graph. This process is not only crucial for rendering graphics but also aids in efficient algorithms for various applications like finite element analysis. By understanding how triangulation works within planar graphs, we can develop better algorithms for computational problems such as mesh generation and optimization.

Related terms

Graph Embedding: The representation of a graph in a geometric space, where vertices correspond to points and edges are represented as curves connecting these points.

Kuratowski's Theorem:

A fundamental theorem stating that a graph is non-planar if and only if it contains a subgraph that is a subdivision of either the complete graph K5 or the complete bipartite graph K3,3.

Face:

A region bounded by edges in a planar graph, including the outer infinite region which is also considered a face.