Exploring Eulerian Paths and Circuits in Undirected Graphs: Finding traversal routes with an even number of edges where each edge is visited exactly once (Eulerian Path) or where the starting and ending vertex are the same (Eulerian Circuit).

Exploring Eulerian Paths and Circuits in Undirected Graphs: Finding traversal routes with an even number of edges where each edge is visited exactly once (Eulerian Path) or where the starting and ending vertex are the same (Eulerian Circuit).

In the realm of graph theory, given an undirected connected graph with nodes and edges, the goal is to write a function that determines whether the graph has an Eulerian circuit or path. An Eulerian graph is one that contains a cycle or path that visits every edge exactly once.

Here's a casual breakdown of the problem:

The given graph is Eulerian if it's possible to draw it without lifting the pen and without retracing any edge. This might sound similar to the Hamiltonian Path problem, but fear not! Finding an Eulerian Path or Cycle is more efficient than solving the Hamiltonian Path problem; it can be achieved in time.

To crack this problem, keep these vital properties of undirected graphs in mind:

1. Eulerian Cycle: A graph has an Eulerian Cycle if and only if:
2. All vertices with non-zero degree belong to a single connected component.
3. Every vertex in the graph has an even degree.
4. Eulerian Path: A graph has an Eulerian Path if and only if:
5. All vertices with non-zero degree belong to the same connected component.
6. Exactly zero or two vertices have odd degree.
7. If zero vertices have odd degree, an Eulerian Cycle exists.
8. If two vertices have odd degree, an Eulerian Path exists.
9. If one vertex has odd degree, it's not possible in an undirected graph.

Here's a straightforward method to implement this approach:

• Create an adjacency list to represent the graph and initialize a visited array for DFS traversal.
• Perform DFS starting from the first vertex with non-zero degree to check the graph's connectivity.
• After DFS, ensure all non-zero degree vertices were visited to confirm the graph is connected.
• Count the number of vertices with odd degree to classify the graph as Eulerian or not.
• Return 2 if the graph has an Eulerian Circuit, 1 if it has an Eulerian Path, and 0 if the graph fails the Eulerian conditions.

WARNING: Graph with no edges is trivially Eulerian. It's Eulerian by definition!

So that's the lowdown on Eulerian Circuits and Paths. Happy traversing!

*Next Article* Introduction to Graph Coloring by kartik

• Graph
• DSA
• Euler-Circuit
• Graph

In the vast field of data-and-cloud-computing and technology, data structures like arrays can be utilized to store the adjacency list for a given graph, allowing for efficient Eulerian Path or Cycle determination. Once we've created this list and performed Depth-First Search (DFS), we can leverage graph theory principles, such as those related to Eulerian Circuits and Paths, to identify the connectivity and odd degree vertices in the graph – properties key to determining whether the graph contains an Eulerian Circuit or Path.

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