# Network Optimization Eulerian Cycles in directed graphs James Orlin 2010

```Network Optimization
Eulerian Cycles in directed
graphs
James Orlin 2010
The initial network
2
1
Every node has
the same
number of arcs
coming in as
going out.
4
5
7
10
6
9
8
11
13
3
12
14
15
2
Determine a tree directed into node 1
2
1
Determine a
bfs (or dfs) tree
directed into
node 1.
4
5
7
In the tree,
each node has
one arc coming
out except
node 1.
10
6
9
8
11
13
3
12
14
15
3
Order the arcs coming out of each node
2
1
The tree arc
out of node j
should be the
last arc of A(j)
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5
7
10
6
9
8
11
13
3
12
14
15
4
Create the walk
2
1
Start at node 1
and create a
walk.
4
5
7
3
6
9
8
Visit nodes of
A(j) in order.
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11
13
12
14
15
5
Continue the walk
2
1
3
Continue the
walk.
4
5
7
6
9
8
Visit nodes of
A(j) in order.
10
11
13
12
14
15
6
Continue the walk
2
1
3
Continue the
walk.
4
5
7
6
9
8
Visit nodes of
A(j) in order.
10
11
13
12
14
15
7
Continue the walk
2
1
3
Continue the
walk.
4
5
7
6
9
8
Visit nodes of
A(j) in order.
10
11
13
12
14
15
8
MITOpenCourseWare
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15.082J / 6.855J / ESD.78J Network Optimization
Fall 2010