Planar Graphs
Graphs drawable without crossing edges.
Key idea
Graphs drawable without crossing edges.
In depth
A graph is planar if it can be drawn in the plane without crossing edges. Euler’s formula V − E + F = 2 holds for connected planar graphs. Kuratowski’s theorem: planar iff it contains no subdivision of K₅ or K₃,₃.
Worked example
K₄ is planar; K₅ is not.
Formula
Real-world uses
- Designing circuit boards and metro maps that avoid crossing wires or lines
Common mistake
Assuming a drawing with crossings means non-planar; it may be redrawn.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Planar Graphs?
Graphs drawable without crossing edges. It is a topic at the expert level in Discrete Math & Number Theory.
What is the formula for Planar Graphs?
The key formula is V − E + F = 2.
Where is Planar Graphs used in real life?
Real-world examples: Designing circuit boards and metro maps that avoid crossing wires or lines.
What should I learn before Planar Graphs?
Make sure you are comfortable with Breadth- and Depth-First Search, the previous topic in Discrete Math & Number Theory.
What is a common mistake with Planar Graphs?
Assuming a drawing with crossings means non-planar; it may be redrawn.
How can I practise Planar Graphs?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Planar Graphs?
Move on to Graph Colouring next.
Related expert topics in Discrete Math & Number Theory
- Strong Induction – Assume all earlier cases to prove the next.
- Well-Ordering Principle – Every nonempty set of positive integers has a least element.
- Partial Orders – Reflexive, antisymmetric, transitive relations.
- Cardinality of Sets – Comparing sizes of infinite sets.
- Countable and Uncountable Sets – ℕ and ℚ are countable; ℝ is not.
- Cantor’s Diagonal Argument – Proof that real numbers are uncountable.