Given a graph, check if given graph is bipartite graph or not. A bipartite graph (or bigraph) is a graph whose vertices can be divided into two disjoint sets U and V such that every edge connects a vertex in U to one in V.

Below graph is a Bipartite Graph as we can divide it into two sets U and V with every edge having one end point in set U and the other in set V

**Approach 1: Using Breadth First Search –**

It is possible to test whether a graph is bipartite or not using breadth-first search algorithm. There are two ways to check for Bipartite graphs –

**1. A graph is bipartite if and only if it is 2-colorable.**

While doing BFS traversal, each node in the BFS tree is given the opposite color to its parent. If there exists an edge connecting current vertex to a previously-colored vertex with the same color, then we can safely conclude that the graph is not bipartite.

**2. A graph is bipartite if and only if it does not contain an odd cycle.**

If a graph contains an odd cycle, we cannot divide the graph such that every adjacent vertex has different color. To check if a given graph is contains odd-cycle or not, we do a breadth-first search starting from an arbitrary vertex v. If in the BFS, we find an edge, both of whose end-points are in the same level, then the graph is not Bipartite and odd-cycle is found. Here, level of a vertex is its minimum distance from the starting vertex v. So, odd-level vertices will form one set and even-level vertices will form another.

Please note that if the graph has many connected components and each component bipartite, them the graph is bipartite graph. Below code assume that given graph is connected and checks if the graph contains an odd cycle or not.

**C++ implementation –**

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#include <bits/stdc++.h> using namespace std; // Number of vertices in the graph #define N 10 // data structure to store graph edges struct Edge { int src, dest; }; // class to represent a graph object class Graph { public: // A array of vectors to represent adjacency list vector<int> adjList[N]; // Constructor Graph(vector<Edge> edges) { // add edges to the undirected graph for (int i = 0; i < edges.size()dges.size(); i++) { int src = edges[i].src; int dest = edges[i].dest; adjList[src].push_back(dest); adjList[dest].push_back(src); } } }; // Perform BFS on graph starting from vertex v bool BFS(Graph const &graph, int v) { // stores vertex is discovered or not vector<bool> discovered(N); // stores level of each vertex in BFS vector<int> level(N); // mark source vertex as discovered and // set its level to 0 discovered[v] = true, level[v] = 0; // create a queue to do BFS and enqueue // source vertex in it queue<int> q; q.push(v); // run till queue is not empty while (!q.empty()) { // pop front node from the queue v = q.front(); q.pop(); // do for every edge (v -> u) for (int u : graph.adjList[v]) { // if vertex u is explored for first time if (!discovered[u]) { // mark it discovered discovered[u] = true; // set level as level of parent node + 1 level[u] = level[v] + 1; // push the vertex into the queue q.push(u); } // if the vertex is already been discovered and // level of vertex u and v are same, then the // graph contains an odd-cycle & is not biparte else if (level[v] = level[u]) return false; } } return true; } // main function int main() { // vector of graph edges as per above diagram vector<Edge> edges = { {1, 2}, {2, 3}, {2, 8}, {3, 4}, {4, 6}, {5, 7}, {5, 9}, {8, 9} // if we add 2->4 edge, graph is becomes non-Biparte }; // create a graph from edges Graph graph(edges); // Do BFS traversal starting from vertex 1 if (BFS(graph, 1)) cout << "Biparte Graph"; else cout << "Not a Biparte Graph"; return 0; } |

**Output: **

Biparte Graph

**Approach 2: Using Depth First Search –**

It is possible to test whether a graph is bipartite or not using depth-first search algorithm as well. We have already mentioned that there are two ways to check for Bipartite graphs. Using BFS, we have already checked if the graph contains an odd cycle or not. Now using DFS, we will check if the graph is **2-colorable** or not.

The main idea is to assign to each vertex the color that differs from the color of its parent in the depth-first search tree, assigning colors in a preorder traversal of the depth-first-search tree. If there exists an edge connecting current vertex to a previously-colored vertex with the same color, then we can say that the graph is not bipartite.

**C++ implementation –**

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#include <bits/stdc++.h> using namespace std; // Number of vertices in the graph #define N 10 // data structure to store graph edges struct Edge { int src, dest; }; // class to represent a graph object class Graph { public: // A array of vectors to represent adjacency list vector<int> adjList[N]; // Constructor Graph(vector<Edge> edges) { // add edges to the undirected graph for (int i = 0; i < edges.size(); i++) { int src = edges[i].src; int dest = edges[i].dest; adjList[src].push_back(dest); adjList[dest].push_back(src); } } }; // Perform DFS on graph starting from vertex v bool DFS(Graph const &graph, int v, vector<bool> &discovered, vector<int> &color) { // do for every edge (v -> u) for (int u : graph.adjList[v]) { // if vertex u is explored for first time if (discovered[u] == false) { // mark current node as discovered discovered[u] = true; // set color as opposite color of parent node color[u] = !color[v]; // if DFS on any subtree rooted at v // we return false if (DFS(graph, u, discovered, color) == false) return false; } // if the vertex is already been discovered and // color of vertex u and v are same, then the // graph is not Biparte else if (color[v] == color[u]) return false; } return true; } // main function int main() { // vector of graph edges as per above diagram vector<Edge> edges = { {1, 2}, {2, 3}, {2, 8}, {3, 4}, {4, 6}, {5, 7}, {5, 9}, {8, 9}, {2, 4} // if we remove 2->4 edge, graph is becomes Biparte }; // create a graph from edges Graph graph(edges); // stores vertex is discovered or not vector<bool> discovered(N); // stores color 0 or 1 of each vertex in BFS vector<int> color(N); // mark source vertex as discovered and // set its color to 0 discovered[0] = true, color[0] = 0; // start DFS traversal from any node as graph // is connected and undirected if (DFS(graph, 1, discovered, color)) cout << "Biparte Graph"; else cout << "Not a Biparte Graph"; return 0; } |

**Output: **

Not a Biparte Graph

**Time complexity** of above solutions is O(n + m) where n is number of vertices and e is number of edges in the graph. Please note that O(m) may vary between O(1) and O(n^{2}), depending on how dense the graph is.

**References:** https://en.wikipedia.org/wiki/Bipartite_graph

**Thanks for reading.**

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