Year 12 Graph Theory: Paths and circuits

Paths and circuits are the movement language of graph theory. A path tells you how to travel through a graph from one vertex to another, while a circuit returns to its starting point. These ideas are basic, but they sit underneath route problems, connectivity arguments, and many network interpretations. The most important habit is to describe the route carefully: which vertices are visited, whether repetition occurs, and whether the start and finish are the same.

A path is about ordered movement through the graph. A circuit is a closed path that returns to where it began.

Subtopic 1: Paths and path length

A path is a sequence of vertices connected by edges in order. In an unweighted graph, the path length is usually the number of edges used. In a weighted graph, the path length is the sum of the edge weights. This distinction is essential because shortest-path questions depend on which notion of length the graph is using.

Worked example 1

Problem: In an unweighted graph, consider the route

A - B - D - E

. What is the path length?

The route uses the edges $A B$ , $B D$ , and $D E$ .
That is a total of three edges.

Answer: the path length is

3

Subtopic 2: Circuits and closed movement

A circuit is a path that starts and ends at the same vertex. Circuits are useful when the problem is about returning to the origin, covering a network repeatedly, or analysing whether certain travel patterns are possible. A route that almost returns but finishes at a different vertex is still a path, not a circuit.

Worked example 2

Problem: Is

A - C - D - A

a circuit?

The route starts at $A$ .
It follows edges through $C$ and $D$ .
It returns to $A$ .

Answer: yes, it is a circuit because it begins and ends at the same vertex.

Why path language matters

Path language looks simple, but it keeps route problems precise. If you only say "there is a way to get there," you lose important information about order, repetition, and length. In graph theory, two routes between the same vertices may have very different consequences: one might be a simple path, another may revisit vertices, and a third may form a circuit. Those distinctions often determine which methods and conclusions are valid.

Worked example 3

Problem: Explain why

A - B - C - B - D

is not a simple path.

A simple path does not repeat vertices.
The route visits $B$ twice.
So the path is valid as a route, but not simple.

Answer: it is not a simple path because vertex

B

is repeated.

From paths to larger graph questions

Understanding paths and circuits prepares you for both connectivity and shortest-path problems. Connectivity asks whether some path exists at all. Shortest path asks which of the possible paths has minimal length. Circuit ideas matter when the route must return to its start or when the graph is being analysed for closed structure. That is why this vocabulary should feel automatic. It is used everywhere in the topic.

Common traps

Calling any route a circuit even when it does not return to its starting point.
Counting vertices instead of edges when measuring an unweighted path length.
Ignoring repeated vertices when deciding whether a path is simple.
Using unweighted logic in a weighted-network question.

Revision focus

In revision, label a few routes on the same graph and classify them as path, simple path, or circuit. This builds much stronger intuition than only reading definitions. Once the language feels natural, more advanced route questions become much easier to decode.

It also helps to write the edges used, not just the vertices visited. That makes path length clearer and prevents ambiguity in crowded diagrams where several different routes may pass through similar-looking regions of the graph.