Planar graph

Definition

A planar graph is a type of graph that can be drawn on a plane without any edges crossing each other. This means that the vertices and edges can be arranged in such a way that no two edges intersect except at their endpoints. Planar graphs have unique properties that make them important in understanding the structure of networks and in solving problems related to connectivity and routing.

5 Must Know Facts For Your Next Test

1. A planar graph can be represented in such a way that no edges intersect, except at their endpoints, allowing for clear visualization.
2. The maximum number of edges in a simple planar graph with V vertices is given by the formula E ≤ 3V - 6 for V ≥ 3.
3. Every connected planar graph can be divided into regions called faces, with each face representing an area enclosed by edges.
4. Planarity testing can be performed using various algorithms, such as the Hopcroft and Tarjan algorithm, which efficiently determines if a graph can be drawn without crossings.
5. In planar graphs, if a graph has more than four vertices, it cannot contain K5 or K3,3 as subgraphs to maintain its planarity.

Review Questions

• How does Euler's Formula apply to planar graphs, and what does it tell us about the relationship between vertices, edges, and faces?
  • 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 relationship provides insight into the structure of planar graphs, indicating that as you add more edges or vertices, you will also affect the number of faces in the graph. It highlights the balance necessary for maintaining planarity while expanding the graph.
• Discuss Kuratowski's Theorem and its significance in determining whether a given graph is planar.
  • Kuratowski's Theorem is crucial because it provides a clear criterion for identifying planar graphs. According to this theorem, a finite graph is planar if it does not contain a subgraph that is a subdivision of K5 or K3,3. This means that by checking for these specific structures within a graph, one can conclusively determine if the graph can be drawn on a plane without crossings. The theorem serves as a foundation for many algorithms used in graph theory.
• Evaluate the implications of planarity in real-world applications, particularly in network design and routing problems.
  • The implications of planarity are significant in real-world applications like network design and routing problems because planar graphs ensure efficient layout with minimal edge crossings. For instance, in telecommunications or transportation networks, planning routes with fewer intersections reduces complexity and potential conflicts. Understanding planarity helps optimize resource allocation and improve connectivity while ensuring safety and efficiency. Thus, recognizing and utilizing planar graphs can lead to more effective solutions in complex logistical scenarios.

Related terms

Euler's Formula: A mathematical equation that relates the number of vertices (V), edges (E), and faces (F) of a connected planar graph: V - E + F = 2.

Kuratowski's Theorem: A fundamental theorem stating that a finite graph is planar if and only if it does not contain a subgraph that is a subdivision of either K5 (the complete graph on five vertices) or K3,3 (the complete bipartite graph on six vertices).

Face: In the context of a planar graph, a face is a region bounded by edges, including the outer, unbounded region.