Walks, Trails, Paths, Cycles and Circuits is a part of the VCE Further Maths topic Networks and Decision Mathematics. It is a part of the subtopic Exploring and Travelling problems.
The topic involves different types of mathematical objects used in graph theory to represent and analyse real-world systems. Graph theory is a branch of mathematics that deals with the study of networks and their properties.
The following definitions are important for Walks, Trails, Paths, Cycles and Circuits:
- Walk: a sequence of edges and vertices in a network. It can start and end at any vertex and can repeat vertices and edges. For example, consider a graph with several interconnected roads, a walk could be a trip starting at one point, going through different roads, and ending at another point, regardless if it passes through the same road more than once.
- Trail: a walk where no edges are repeated. In the previous example of roads, a trail would be a trip that starts and ends at the same point, but it never takes the same road twice.
- Path: a trail that connects all vertices. In the roads example, a path would be a trip that starts and ends at the same point, it never takes the same road twice, and never passes through the same intersection more than once.
- Cycle: a trail that finishes on the vertex it begins on. No vertex is repeated.
- Circuit: a path that begins and ends on the same vertex. Vertices can be repeated.
Network Travel Terminology
The following video introduces the terminology involved in network travel including; walks, trails, paths, circuits.
Identifying the walk
The following video explains how to identify which type of walk a network undertakes using the appropriate. terminology
Traversable and Non-Traversable Graphs
A traversable graph has a trail that includes every edge. A non-traversable graph has a trail that does not include every edge. The following video explains the difference between traversable and non-traversable graphs.
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