Discrete Mathematics
Fleury’s Algorithm
Eulerization of Graphs
Name: ________________________
Date: ______________________
We practiced finding Euler Circuits and Euler Paths in graphs that were reasonably sized. What if you had to find
one in a larger, more complex graph? We can use ________________________________________________________
Definition: Algorithm
Fleury’s Algorithm for Finding an Euler Circuit (Path)
 Preliminaries –

Start –

Intermediate steps –

End –
A
A
B
B
C
E
D
C
F
E
D
F
Question: If a graph doesn’t have an Euler path or circuit, how can we modify the graph so that is does?
Answer: EULERIZATION and SEMI-EULERIZATION!!
EULERIZATION
If a graph does not have an Euler circuit, we still might be interested in knowing how it could be traveled with as
few retraced edges as possible (starting and ending at the same vertex). These retraced edges can be indicated by
adding duplicate edges between two vertices to indicate that an edge between those two vertices was retraced.
Although these duplicate edges just represent a retracing of an edge rather than an actual new edge,
mathematically it doesn’t matter whether they are real or not.
This process is called _____________________________________________________________.
Eulerizing a Graph
Notes about Eulerization of a graph:
1. The __________________________________________ added the to Eulerized graph can be thought of as actual new edges
but in many contexts, the new edges of the Eulerized graph just represent a retracing of the same edge in the
original graph.
2. Eulerization only involves adding edges which duplicate existing edges. You _______________________ add “new”
edges; that is, edges which do not duplicate edges already in the graph.
3. Always ____________________________ a graph using the ______________________________ edges possible.
4. The duplicate edges added to a graph during the process of Eulerization are often called
_____________________________________________
Eulerize the following graphs.
1.
2.
3.
SEMI-EULERIZATION
The only difference between EULERIZATION and SEIM-ELUERIZATION is that eulerization results in an Euler
circuit – with the same starting and ending points – while semi-eulerization results in an Euler path – with
different starting and ending points.
Semi-Eulerizing a Graph
The process of adding duplicate edges to a graph so that the resulting graph has
_____________________________________________________________________ (and thus
does have an Euler path but not an Euler circuit) is called
__________________________________
Semi-eulerize this graph.
4.