Non-planar graph

Definition

A non-planar graph is a type of graph that cannot be drawn on a flat plane without edges crossing. This means that no matter how you arrange the vertices and edges, at least one pair of edges will overlap in a way that violates the rules of planar graphs. Non-planar graphs often have complex structures and are significant in understanding certain properties and behaviors in geometric graph theory as well as in planarity testing and embedding.

5 Must Know Facts For Your Next Test

1. The most famous examples of non-planar graphs are K5 (the complete graph on five vertices) and K3,3 (the complete bipartite graph with two sets of three vertices).
2. Non-planar graphs can be identified using planarity testing algorithms, such as the Hopcroft and Tarjan algorithm, which efficiently determine whether a given graph can be embedded in the plane.
3. The study of non-planar graphs is crucial in various applications, including network design, circuit layout, and geographical mapping, where real-world constraints often lead to non-planarity.
4. Non-planarity can lead to complications in visualizing graphs, making it essential to use alternative representations or methods for understanding their structure and properties.
5. Understanding non-planar graphs aids in solving problems related to colorings and coverings in graph theory, since certain properties may differ significantly from those of planar graphs.

Review Questions

• What characteristics distinguish a non-planar graph from a planar graph?
  • Non-planar graphs cannot be drawn on a flat surface without edge crossings, which is the key characteristic that sets them apart from planar graphs. In contrast, planar graphs can be represented in such a way that no edges intersect each other. This distinction leads to different properties and behaviors between the two types of graphs, influencing how they are analyzed and utilized in various applications.
• How does Kuratowski's Theorem help determine whether a graph is non-planar?
  • Kuratowski's Theorem provides a clear criterion for identifying non-planar graphs by stating that a graph is non-planar if it contains a subgraph that is either K5 or K3,3. This means that if you can find these specific configurations within any given graph, you can definitively conclude it is non-planar. This theorem plays an essential role in both theoretical studies of graph properties and practical applications where planarity matters.
• Evaluate the implications of non-planarity on practical applications such as network design or circuit layout.
  • Non-planarity has significant implications in fields like network design and circuit layout because real-world systems often can't be represented as planar graphs due to physical constraints. In these situations, understanding non-planarity helps engineers optimize designs to reduce edge crossings or manage complex interconnections effectively. The challenges posed by non-planarity necessitate the use of specialized algorithms for testing and embedding, ultimately affecting efficiency and functionality in practical implementations.

Related terms

Planar Graph: A planar graph is a graph that can be drawn on a plane without any of its edges crossing each other.

Kuratowski's Theorem: A fundamental theorem in graph theory that states a 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.

Graph Embedding: The representation of a graph in a topological space, such as a plane or sphere, where vertices are points and edges are curves connecting them, aiming for minimal edge crossings.