Planar graphs

Definition

A planar graph is a graph that can be drawn on a plane without any of its edges crossing. This property is crucial for understanding the relationship between graph theory and polyhedra, as it helps in visualizing and analyzing the structure of various geometric shapes. Planar graphs adhere to specific rules, such as Euler's formula, which connects the number of vertices, edges, and faces in a way that reveals significant insights into their topology.

5 Must Know Facts For Your Next Test

1. Planar graphs can be embedded in a two-dimensional plane without any edge crossings, which makes them easier to visualize.
2. Euler's formula, V - E + F = 2, holds true for connected planar graphs, helping to establish a fundamental relationship between vertices, edges, and faces.
3. A common application of planar graphs is in network design and circuit layout, where minimizing intersections is critical for efficiency.
4. Kuratowski's theorem provides a method to determine whether a graph is planar by checking for certain subdivisions that would indicate non-planarity.
5. Many polyhedra can be represented as planar graphs, where vertices correspond to corners, edges correspond to edges of the shape, and faces correspond to the surfaces.

Review Questions

• How does Euler's formula relate to planar graphs and what does it reveal about their structure?
  • Euler's formula states that for any connected planar graph, the equation V - E + F = 2 holds true, where V is the number of vertices, E is the number of edges, and F is the number of faces. This formula reveals a fundamental relationship among these three components and helps identify characteristics of planar graphs. For example, it can assist in determining whether a graph can be drawn without crossings by providing insight into how changes in one component affect the others.
• What does Kuratowski's theorem tell us about identifying non-planar graphs?
  • Kuratowski's theorem states that a finite graph is planar if and only if it does not contain a subgraph that is a subdivision of K5 (the complete graph on five vertices) or K3,3 (the complete bipartite graph). This theorem provides a clear method for assessing planarity; if such subdivisions are present within a graph, it confirms that the graph cannot be drawn without edge crossings. Therefore, Kuratowski's theorem serves as an important tool for determining planarity in more complex graphs.
• Evaluate the implications of dual graphs in relation to planar graphs and polyhedra.
  • The concept of dual graphs plays a crucial role in understanding the relationship between planar graphs and polyhedra. In a dual graph, each vertex corresponds to a face of the original planar graph, while edges represent adjacency between these faces. This duality provides insights into properties like connectivity and helps in applying algorithms related to optimization problems. By studying dual graphs, one can explore various characteristics of planar structures and gain a deeper understanding of their topological properties.

Related terms

Euler's formula: A formula that states for any convex polyhedron, the relationship between the number of vertices (V), edges (E), and faces (F) is given by the equation V - E + F = 2.

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

Dual graph: A graph that corresponds to another graph in which vertices represent the faces of the original graph and edges represent the adjacency between those faces.