Eulerian path and circuit for undirected graph

Given an undirected connected graph with v nodes, and e edges, with adjacency list adj. We need to write a function that returns 2 if the graph contains an eulerian circuit or cycle, else if the graph contains an eulerian path, returns 1, otherwise, returns 0.

A graph is said to be Eulerian if it contains an Eulerian Cycle, a cycle that visits every edge exactly once and starts and ends at the same vertex.
If a graph contains an Eulerian Path, a path that visits every edge exactly once but starts and ends at different vertices, it is called Semi-Eulerian.

Examples:

Input:

👁 Euler1

Output: 1

Input:

👁 Euler2

Output: 2

Input:

👁 Euler3

Output: 0

The problem can be framed as follows: "Is it possible to draw a graph without lifting your pencil from the paper and without retracing any edge?"
At first glance, this may seem similar to the Hamiltonian Path problem, which is NP-complete for general graphs. However, determining whether a graph has an Eulerian Path or Cycle is much more efficient: It can be solved in O(v + e) time.

Approach:

The idea is to use some key properties of undirected graphs that help determine whether they are Eulerian (i.e., contain an Eulerian Path or Cycle) or not.

Eulerian Cycle

A graph has an Eulerian Cycle if and only if the below two conditions are true

• All vertices with non-zero degree are part of a single connected component. (We ignore isolated vertices— those with zero degree — as they do not affect the cycle.)
• Every vertex in the graph has an even degree.

Eulerian Path

A graph has an Eulerian Path if and only if the below two conditions are true

• All vertices with non-zero degree must belong to the same connected component. (Same as Eulerian Cycle)
• Exactly Zero or Two Vertices with Odd Degree:
  • If zero vertices have odd degree → Eulerian Cycle exists (which is also a path).
  • If two vertices have odd degree → Eulerian Path exists (but not a cycle).
  • If one vertex has odd degree → Not possible in an undirected graph. (Because the sum of all degrees in an undirected graph is always even.)

Note: A graph with no edges is trivially Eulerian. There are no edges to traverse, so by definition, it satisfies both Eulerian Path and Cycle conditions.

How Does This Work?

• In an Eulerian Path, whenever we enter a vertex (except start and end), we must also leave it. So all intermediate vertices must have even degree.
• In an Eulerian Cycle, since we start and end at the same vertex, every vertex must have even degree. This ensures that every entry into a vertex can be paired with an exit.

Steps to implement the above idea:

• Create an adjacency list to represent the graph and initialize a visited array for DFS traversal.
• Perform DFS starting from the first vertex having non-zero degree to check graph 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.
• If all degrees are even, the graph has an Eulerian Circuit; if exactly two are odd, it's a Path.
• If more than two vertices have odd degree or graph isn't connected, it's not Eulerian.
• Return 2 for Circuit, 1 for Path, and 0 when graph fails Eulerian conditions.

1

Time Complexity: O(v + e), DFS traverses all vertices and edges to check connectivity and degree.
Space Complexity: O(v + e), Space used for visited array and adjacency list to represent graph.