Planar graph

Definition

A planar graph is a graph that can be drawn on a flat surface without any of its edges crossing each other. This property allows for visual representations of relationships between vertices in a two-dimensional space, making it useful in various applications such as geographic information systems and circuit design. Planar graphs also have important properties related to their structure, such as Euler's formula, which relates the number of vertices, edges, and faces in a connected planar graph.

5 Must Know Facts For Your Next Test

1. A planar graph can be drawn without edge crossings, which is crucial for clear visualization of relationships.
2. There are limits on how many edges a planar graph can have; specifically, for a connected planar graph with V vertices, it can have at most 3V - 6 edges when V >= 3.
3. Not all graphs are planar; determining whether a graph can be drawn without crossings is essential in many applications.
4. Planar graphs are closely related to polyhedra in three dimensions, where each face corresponds to an edge in the graph.
5. The dual of a planar graph is another planar graph formed by associating each face of the original graph with a vertex in the dual.

Review Questions

• How does Euler's formula apply to planar graphs, and what significance does it hold?
  • Euler's formula states that for any connected planar graph, the relationship between vertices (V), edges (E), and faces (F) is given by V - E + F = 2. This formula helps us understand the fundamental structure of planar graphs and provides insight into how adding or removing elements affects the overall connectivity. It serves as a powerful tool for verifying whether a given drawing of a graph maintains planarity.
• What is Kuratowski's theorem, and how does it help in determining if a graph is planar?
  • Kuratowski's theorem provides a criterion for identifying planar graphs by stating that a graph is planar if it does not contain a subgraph that is a subdivision of either K5 (the complete graph on five vertices) or K3,3 (the complete bipartite graph). This theorem is significant because it gives us specific configurations to look for when assessing planarity, allowing for systematic testing of complex graphs.
• Evaluate the implications of having non-planar graphs in real-world applications, particularly in circuit design or geographic information systems.
  • Non-planar graphs can lead to complications in real-world applications like circuit design or geographic information systems because they may create overlapping connections that are difficult to manage. For instance, in circuit design, overlapping wires can cause short circuits or inefficiencies. Recognizing when a design leads to non-planarity allows engineers and designers to rethink their layouts to ensure functionality and clarity. Thus, understanding planarity plays a critical role in optimizing designs and preventing errors.

Related terms

Euler's formula: A formula that relates the number of vertices (V), edges (E), and faces (F) in a connected planar graph, expressed as V - E + F = 2.

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

face: A face in a planar graph is a region bounded by edges, including the outer infinite region in the drawing of the graph.