Paths, Cycles and Connectivity
Walks through graphs and connected components.
Key idea
Walks through graphs and connected components.
In depth
A path is a sequence of edges without repeating vertices; a cycle is a closed path. A graph is connected if a path exists between any two vertices. Components are maximal connected pieces.
Worked example
A graph of two separate triangles has 2 components.
Real-world uses
- Checking whether every town in a network can be reached
- Finding loops in a circuit
Common mistake
Confusing a walk (can repeat vertices) with a path.
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 Paths, Cycles and Connectivity?
Walks through graphs and connected components. It is a topic at the advanced level in Discrete Math & Number Theory.
Where is Paths, Cycles and Connectivity used in real life?
Real-world examples: Checking whether every town in a network can be reached; Finding loops in a circuit.
What should I learn before Paths, Cycles and Connectivity?
Make sure you are comfortable with Types of Graphs, the previous topic in Discrete Math & Number Theory.
What is a common mistake with Paths, Cycles and Connectivity?
Confusing a walk (can repeat vertices) with a path.
How can I practise Paths, Cycles and Connectivity?
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 Paths, Cycles and Connectivity?
Move on to Euler Paths and Circuits next.
Related advanced topics in Discrete Math & Number Theory
- Tautologies and Contradictions – Always true or always false statements.
- Logical Equivalence – Statements with identical truth tables.
- De Morgan’s Laws – Negation swaps AND and OR.
- Converse and Contrapositive – A conditional equals its contrapositive.
- Quantifiers – “For all” and “there exists”.
- Negating Quantified Statements – Flip the quantifier and negate the predicate.