Planar graph

Definition

A planar graph is a graph that can be drawn on a flat surface without any edges crossing each other. This property allows planar graphs to be visually represented in a way that makes it easy to analyze their structure and relationships. The concept of planarity is essential in various fields, including computer graphics and geographical mapping, as it helps simplify complex networks.

5 Must Know Facts For Your Next Test

1. Planar graphs can have at most 3V - 6 edges for graphs with V ≥ 3, which is derived from Euler's Formula.
2. Every planar graph can be colored using at most four colors without any two adjacent vertices sharing the same color, known as the Four Color Theorem.
3. A common method for testing if a graph is planar is through embedding algorithms or by attempting to draw the graph while avoiding edge crossings.
4. The presence of certain subgraphs, like K5 or K3,3, indicates that a graph is non-planar based on Kuratowski's Theorem.
5. Planar graphs are widely used in applications like circuit design, network topology, and geographical information systems due to their structural simplicity.

Review Questions

• How can Euler's Formula be used to determine properties of a planar graph?
  • Euler's Formula relates the number of vertices, edges, and faces in a connected planar graph through the equation V - E + F = 2. By knowing any two of these values, you can derive the third. For example, if you have a planar graph with 5 vertices and 7 edges, you can use this formula to find that it must have 4 faces. This relationship helps in analyzing the structure and complexity of planar graphs.
• In what ways does Kuratowski's Theorem help in identifying non-planar graphs?
  • Kuratowski's Theorem states that a finite graph is non-planar if it contains a subdivision of either K5 or K3,3. This provides a clear criterion for determining planarity. By examining a graph for these specific configurations, you can quickly ascertain whether it's possible to draw the graph without edge crossings. This theorem serves as a foundational tool in graph theory for studying planarity.
• Discuss the implications of the Four Color Theorem in relation to planar graphs and its applications.
  • The Four Color Theorem states that any planar graph can be colored with no more than four colors such that no adjacent vertices share the same color. This has significant implications in various fields such as map coloring, where different regions must be represented distinctly without overlap. The theorem not only enhances our understanding of planar graphs but also has practical applications in scheduling problems and network design, where avoiding conflicts is crucial.

Related terms

Euler's Formula: A formula that establishes a relationship between the number of vertices (V), edges (E), and faces (F) in a connected planar graph, given by the equation V - E + F = 2.

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

Graph Coloring: The process of assigning colors to the vertices of a graph such that no two adjacent vertices share the same color, which is particularly relevant in the context of planar graphs.