Traversable networks
If a network has a trail or circuit that uses every edge, it is described as 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).
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:
Each network has a trail that uses every edge exactly once. The trail on the right is closed, so it is a circuit, and any vertex can be the start/end
Practice questions
Question 1
Is the network below traversable?
No
AYes
B
Question 2
Is the network below traversable?
No
AYes
B
Question 3
Consider the networks below:
Which one of the networks is traversable?
AB
C
DThe traversable network from part (a) has been labelled below:
Which vertices can you start at in order to traverse the network?
Select the most correct answer only.
$A$A, $C$C and $G$G.
A$D$D and $F$F.
BJust $C$C.
C$B$B and $E$E.
DAny of the vertices.
EJust $D$D.
F