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C++ Program to Check Whether an Undirected Graph Contains a Eulerian Path
The Euler path is a path; by which we can visit every node exactly once. We can use the same edges for multiple times. The Euler Circuit is a special type of Euler path. When the starting vertex of the Euler path is also connected with the ending vertex of that path.
To detect the Euler Path, we have to follow these conditions
- The graph must be connected.
- Now when no vertices of an undirected graph have odd degree, then it is a Euler Circuit, which is also one Euler path.
- When exactly two vertices have odd degree, it is a Euler Path.
Input
Output
Both of the graphs has Euler paths.
Algorithm
traverse(u, visited)
Input : The start node u and the visited node to mark which node is visited.
Output : Traverse all connected vertices.
Begin mark u as visited for all vertex v, if it is adjacent with u, do if v is not visited, then traverse(v, visited) done End
isConnected(graph)
Input : The graph.
Output : True if the graph is connected.
Begin define visited array for all vertices u in the graph, do make all nodes unvisited traverse(u, visited) if any unvisited node is still remaining, then return false done return true End
isEulerian(Graph)
Input : The given Graph.
Output : Returns 1, when Eulerian circuit or path, and returns 0 when it has no Euler Path.
Begin if isConnected() is false, then return false define list of degree for each node oddDegree := 0 for all vertex i in the graph, do for all vertex j which are connected with i, do increase degree done if degree of vertex i is odd, then increase oddDegree done if oddDegree > 0, then return 0 else return 1 End
Example Code
#include<iostream>
#include<vector>
#define NODE 5
using namespace std;
int graph[NODE][NODE] = {{0, 1, 1, 1, 0},
{1, 0, 1, 0, 0},
{1, 1, 0, 0, 0},
{1, 0, 0, 0, 1},
{0, 0, 0, 1, 0}};
/*int graph[NODE][NODE] = {{0, 1, 1, 1, 1},
{1, 0, 1, 0, 0},
{1, 1, 0, 0, 0},
{1, 0, 0, 0, 1},
{1, 0, 0, 1, 0}};*/ //uncomment to check Euler Circuit as well as path
/*int graph[NODE][NODE] = {{0, 1, 1, 1, 0},
{1, 0, 1, 1, 0},
{1, 1, 0, 0, 0},
{1, 1, 0, 0, 1},
{0, 0, 0, 1, 0}};*/ //Uncomment to check Non Eulerian Graph
void traverse(int u, bool visited[]) {
visited[u] = true; //mark v as visited
for(int v = 0; v<NODE; v++) {
if(graph[u][v]) {
if(!visited[v])
traverse(v, visited);
}
}
}
bool isConnected() {
bool *vis = new bool[NODE];
//for all vertex u as start point, check whether all nodes are visible or not
for(int u; u < NODE; u++) {
for(int i = 0; i<NODE; i++)
vis[i] = false; //initialize as no node is visited
traverse(u, vis);
for(int i = 0; i<NODE; i++){
if(!vis[i]) //if there is a node, not visited by traversal, graph is not connected
return false;
}
}
return true;
}
int isEulerian() {
if(isConnected() == false) //when graph is not connected
return 0;
vector<int> degree(NODE, 0);
int oddDegree = 0;
for(int i = 0; i<NODE; i++) {
for(int j = 0; j<NODE; j++) {
if(graph[i][j])
degree[i]++; //increase degree, when connected edge found
}
if(degree[i] % 2 != 0) //when degree of vertices are odd
oddDegree++; //count odd degree vertices
}
if(oddDegree > 2) //when vertices with odd degree greater than 2
return 0;
return 1; //when oddDegree is 0, it is Euler circuit, and when 2, it is Euler path
}
int main() {
if(isEulerian() != 0) {
cout << "The graph has Eulerian path." << endl;
} else {
cout << "The graph has No Eulerian path." << endl;
}
}Output
The graph has Eulerian path.
Updated on: 30-Jul-2019
672 Views
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