Question

Devise an algorithm to find an Eulerian circuit in a directed graph, if one exists. Modify the algorithm to find all Eulerian circuits in a graph.

Short answer

Use Hierholzer's Algorithm to find one circuit, and extend it recursively to find all.

Step-by-step solution

Understanding Eulerian Circuit

An Eulerian circuit in a directed graph is a cycle that visits every edge exactly once and returns to the starting vertex. To have an Eulerian circuit, each vertex's in-degree must equal its out-degree, and all vertices with non-zero degree must be strongly connected.

Verify Graph Conditions for Eulerian Circuit

Verify that the graph meets Eulerian circuit conditions: for every vertex, check if the in-degree equals the out-degree, and ensure all non-zero degree vertices are in the same strongly connected component via depth-first search or another approach.

Use Hierholzer’s Algorithm for Eulerian Circuit

Start finding the Eulerian circuit using Hierholzer’s Algorithm. Begin with any vertex having outgoing edges and keep following edges until returning to the starting vertex, forming a circuit. Remove used edges from the sequence.

Extend Hierholzer's to Find All Circuits

To find all Eulerian circuits, iterate through each circuit generated. For vertices in this circuit with untraversed outgoing edges, recursively use Hierholzer’s to find and integrate new circuits where possible.

Integrate and Output All Circuits

As recursive calls reach completion, integrate all found circuits into separate distinct circuits and output them. Each distinct integrated path forms one complete Eulerian circuit in the graph.

Key concepts

Directed Graph

A directed graph, often referred to as a digraph, is a set of vertices connected by edges where the edges have a direction. This means each edge has a start vertex and an end vertex, indicating the direction from one vertex to the other.

• Components: Directed graphs consist of vertices (or nodes) and directed edges.
• Representation: They can be represented using adjacency lists or matrices, making it easy to identify edge directions.
• Applications: Directed graphs are widely used in real-world scenarios like modeling websites for page ranking or analyzing routes in traffic networks.

In the context of Eulerian circuits, directed graphs require specific conditions regarding in-degrees and out-degrees of vertices to form such circuits.

Hierholzer’s Algorithm

Hierholzer’s algorithm is an efficient method to find an Eulerian circuit in a graph, provided certain conditions are met. Its approach is intuitive for tracing paths within a graph.
The algorithm initiates from any vertex with outgoing edges and constructs a cycle by continually following directed edges until returning to the starting vertex. Importantly, it removes edges as they are followed to prevent re-visiting in subsequent cycles.
To handle disconnected parts within the path, the algorithm integrates newly discovered cycles by splicing them into already found cycles. This powerful additive blending allows the entire Eulerian path to emerge as more edges are traversed and cycles connected.

Strongly Connected Component

A strongly connected component (SCC) in a directed graph is a subset of vertices where every vertex is reachable from every other vertex in the subset. To determine if a graph is strongly connected, one can use depth-first search (DFS) or algorithms like Kosaraju's and Tarjan's.
For a directed graph to contain an Eulerian circuit, it is vital that all vertices with outbound edges belong to the same SCC. This ensures there are no isolated paths, facilitating a continuous cycle spanning every edge.
Identifying SCCs:

• Use depth-first search (DFS) to navigate and mark traversed vertices.
• Kosaraju's Algorithm performs two passes with DFS to identify SCCs efficiently.
• Tarjan's Algorithm finds SCCs using DFS with a single linear pass.

Understanding strongly connected components ensures correct Eulerian circuit formation in a directed graph.

In-degree and Out-degree

In a directed graph, the in-degree of a vertex is the number of edges pointing into it, while the out-degree is the number pointing out. Balancing these degrees is crucial for forming an Eulerian circuit.
For an Eulerian circuit to exist, every vertex must have its in-degree equal to its out-degree. This means every time a path exits a vertex, there is an equal path entering, allowing seamless transitions through the graph. Significance in Eulerian Circuits:

• Verifying degree equality ensures feasible round trips through vertices.
• If any vertex has differing in and out degrees, the graph cannot have a complete Eulerian circuit.
• This condition checks balance, making Eulerian traversal possible without missing any edges.

Checking in-degree and out-degree is a preliminary step before attempting to find Eulerian circuits using algorithms like Hierholzer’s.