Hierholzer's algorithm is designed to find an Eulerian circuit in a graph by following specific steps to traverse edges and merge cycles. The algorithm is efficient and simple to implement, making it suitable for both directed and undirected graphs. The document includes an overview of the algorithm, steps for coding it in Python, and the benefits of using Hierholzer's for solving Eulerian circuit problems.

1. HIERHOLZER'
S
ALGORITHM
Graph Theory
1

2. OVERVIEW
2
• Purpose
• Steps of Algo
• Code using Python
• Steps of Coding
• Why Hierholzer's
2

3. PURPOSE
1
2
3
Hierholzer's Algorithm is used to find an Eulerian Circuit in
a graph.
The output is a list of vertices representing the sequence of
nodes in the Eulerian Circuit.
Hierholzer’s Algorithm is faster, simpler, and purpose-built for
Eulerian Circuits, making it the best choice for this specific
problem.
3
3

4. STEPS OF HIERHOLZER'S ALGO:
3
4
1. Start at any vertex and follow edges until you return to
the starting vertex, forming a cycle.
2. If the cycle doesn't cover all edges, choose a vertex on the
cycle with unused edges and form a new cycle.
3. Merge the cycles into a single Eulerian circuit.
4. Repeat until all edges are used.

5. EXAMPLE:
3
5

6. CODE IN PYTHON 4
6

7. OUTPUT
5
7

8. STEPS OF CODING
6
The graph is
represented as a
dictionary where
each key is a node
and its value is a list
of connected
neighbors.
1.Input
Graph: The algorithm
works on a copy
of the graph to
ensure we don't
modify the
original graph.
2. Copy Graph:
8

9. 7
• Use a current_path
stack to keep track
of the nodes being
traversed.
• Use a circuit list to
store the final
Eulerian Circuit.
3. Path
Tracking: • Pick a starting node
and traverse until
all edges are visited.
4. Algorithm
Steps:
• If a node has no
neighbors, backtrack
to a previous node
with unused edges.
9

10. 8
• The Eulerian circuit
is returned as a list
of nodes.
5. Output:
10

11. 9
WHY HIERHOLZER'S
1.Specialized: Optimized for Eulerian circuits, handling specific conditions
efficiently.
2. Efficient: visiting each edge exactly once.
3. Simple: Easy to implement with stacks/lists, avoiding recursion overhead.
4. Versatile: Works for both undirected and directed graphs.
5. Guaranteed Output: Always finds a circuit if Eulerian conditions are met.
Why Hierholzer's Algorithm is Better

12. THANK YOU
ANY QUESTION?
12