Introduction to Planar Graphs

Overview

• Focus on planar graphs, with an introduction to equivalent graphs (isomorphic graphs).
• Definitions and visual examples provided.

Equivalent Graphs (Isomorphic Graphs)

• Graphs that have:
  • The same vertices.
  • The same edges.
  • Same connections between vertices and edges.
• Can be drawn without edges crossing over, making them more visually pleasing.

Planar Graphs

• Definition: A graph that can be drawn so that no edges cross over.
• Example: Graphs that are equivalent and can be rearranged to have no crossing edges are planar.

Non-Planar Graphs

• Definition: These graphs cannot be drawn in any way without edges crossing over.
• Example: A graph with many vertices and edges that cannot be rearranged to avoid crossing.

Planar Form

• Planar Form: The visual representation of a planar graph where no edges cross.
• A graph can be a planar graph if it can be drawn in planar form, even if initially presented with crossing edges.

Key Points

• Planar graphs can be redrawn to eliminate crossing edges, achieving planar form.
• Non-planar graphs cannot be redrawn in such a way, meaning edges will always cross.

Importance

• Understanding the distinction between planar and non-planar graphs is crucial for graph theory studies.
• Upcoming discussions will explore more complex aspects of planar graphs.