Networks

NZ Level 7 (NZC) Level 2 (NCEA)

Traversable networks

Lesson

If a network has a trail or circuit that uses every edge, we call the network **traversable** (sometimes **semi-Eulerian**). This special walk is then called an **Euler trail*** if it’s open or an **Euler circuit** if it’s closed, named after the same Euler that gave us Euler’s formula. If a network has an Euler circuit we call it an **Euler network** (sometimes **Eulerian**).

*Watch out!

Some texts refer to such a trail as an **Euler path**, but this special walk is a **trail** (since vertices can be repeated), *not* a **path**!

You can think about a traversable networks as ones where you can start at a vertex and draw in all the edges without taking your pen (or finger or stylus) off the page (or screen). Here are two traversable networks, the one on the right is an Euler network:

We will discover a way to find these trails and circuits in a later lesson.

Choose appropriate networks to find optimal solutions

Apply network methods in solving problems