THEORY
An Euler path, in a graph or multigraph, is a walk through the graph which uses every edge exactly once. It starts and ends at different vertices.
Suppose that a graph has an Euler path P. For every vertex v other than the starting and ending vertices, the path P enters v the same number of times that it leaves v (say n times). Therefore, there are 2n edges having v as an endpoint.
Therefore, all vertices other than the two endpoints of P must be even vertices. Therefore, if a graph G has an Euler path, then it must have exactly two odd vertices.
An Euler circuit is a circuit that uses every edge of a graph exactly once. It starts and ends at the same vertex. Suppose that a graph G has an Euler circuit C. For every vertex v in G, each edge having v as an endpoint shows up exactly once in C. The circuit C enters v the same number of times that it leaves v (say n times), so v has degree 2n. That is, v must be an even vertex. Therefore, if a graph G has an Euler circuit, then all of its vertices must be even vertices.
EXAMPLE 1
GRAPH:
All vertices of non-zero degree in graph-1 are “Connected” & total number of vertices with odd degrees are “Zero” hence the given graph-1 is “Eulerian Cycle”. 3 -> 1 -> 2 -> 4 -> 3
Similar for the graph-2 all the vertices with non-zero degrees are connected and vertices with odd degrees are “Zero” hence graph-2 is also “Eulerian Cycle”. 2 -> 1 -> 3 -> 4 -> 5 -> 6 -> 3 -> 2
EXAMPLE 2
GRAPH:
All the vertices with non-zero degrees are connected and total number of vertices with odd degrees are “Two” hence the graph is “Eulerian Path” Path: 4 -> 3 -> 0 -> 1 -> 2 -> 0
ALGORITHM
BASIC IDEA
Eulerian or Eulerian Cycle
• First step is to check if all the Vertices with “Non-Zero” Degree are “Connected” or not (Vertices with zero degrees doesn`t belong to Eulerian Cycle)
• Second step is to count the number of vertices with “ODD” degree. If the count is “Zero” then the graph is “EULERIAN OR EULERIAN CYCLE”.
Semi-Eulerian or Eulerian Path
• First step is similar to Eulerian.
• Second step is to count the number of vertices with “ODD” degree. If the count is “Two” then the graph is “SEMI-EULERIAN OR EULERIAN PATH”.
EXPLAINATION
INPUT
- Number of Vertices (4)
- Edges (Ex. 1 2)
STEP BY STEP EXPLAINATION OF CODE
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Creating Adjacency list of all vertices in graph. (Ex. Head[1]- > 2 -> 4 Head[2]-> 1 -> 4 -> 3 Head[3]-> 2 -> 4 Head[4]-> 1 -> 2 -> 3 )
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Finding first vertex with “Non-Zero” to start traversal. (Here 1 is first non-zero degree vertex) (To Visit every Vertex which is connected to this Vertex, by this we can check if all vertices with non-zero degree are connected or not) (Corner Case: If graph does not contain any edges in it then it will be “Eulerian”-Because there are no edges to traverse)
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After traversing through graph, check if all vertices with non-zero degree are visited. If not then the given graph will not be “Eulerian or Semi-Eulerian” And Code will end here. (Here in given example all vertices with non-zero degree are visited hence moving further).
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After passing step 3 correctly -> Counting vertices with “ODD” degree. If count is “ZERO” then the graph is “Eulerian or Eulerian Cycle”. If count is “TWO” then the graph is “Semi-Eulerian or Eulerian Path” . If count is “More than Two” then the graph is “Not-Eulerian”.
OUTPUT
For the given Example output will be -> “Eulerian Path” (Odd degree vertices are two 2&4).