Proper embedding

Definition

Proper embedding refers to the representation of a graph in the plane such that no edges intersect except at their endpoints. This concept is crucial for visualizing graphs clearly without any confusion caused by edge crossings, ensuring that each vertex is represented by a distinct point in the plane. Proper embeddings help in determining the planarity of a graph, which indicates whether it can be drawn on a flat surface without overlaps.

5 Must Know Facts For Your Next Test

1. Proper embedding ensures that each vertex is uniquely identifiable by a point in the plane, which enhances clarity in visualizations.
2. If a graph has a proper embedding, it is classified as planar and can be drawn without edge intersections aside from at shared vertices.
3. The concept of proper embedding is vital for algorithms designed to test graph planarity, allowing them to determine if such an embedding exists.
4. Proper embeddings can reveal properties about graph structures, such as the number of faces formed in a planar drawing.
5. While creating a proper embedding, care must be taken to avoid crossing edges, as this would violate the definition and affect planarity.

Review Questions

• What criteria must be met for a graph to be considered as having a proper embedding?
  • For a graph to have a proper embedding, it must be represented in such a way that no two edges intersect except at their endpoints. This means every vertex must be represented by a unique point in the plane, and edges must connect these points without crossing over each other. If these criteria are satisfied, the graph can be considered planar.
• How does Kuratowski's theorem relate to proper embeddings and planarity testing?
  • Kuratowski's theorem provides a foundational basis for understanding planarity and proper embeddings. It states that a graph is non-planar if it contains a subgraph that is a subdivision of either K5 or K3,3. This theorem aids in planarity testing by providing criteria to identify non-planar graphs, which cannot have proper embeddings without edge crossings.
• Evaluate the significance of proper embeddings in practical applications such as network design or circuit layout.
  • Proper embeddings play an essential role in practical applications like network design and circuit layout because they ensure clarity and efficiency in visual representation. In networking, minimizing edge crossings helps reduce complexity and errors in routing paths. Similarly, in circuit design, proper embeddings assist in laying out components without overlaps, which can lead to better performance and easier troubleshooting. Understanding how to achieve proper embeddings allows engineers and designers to optimize their designs for functionality and aesthetics.

Related terms

Planar graph: A graph that can be drawn on a plane without any edges crossing.

Kuratowski's theorem: A theorem stating that a finite 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 of the plane bounded by edges in a proper embedding of a planar graph, including the outer infinite region.