Is Graph Bipartite?

Problem

There is an undirected graph with n nodes, where each node is numbered between 0 and n - 1. You are given a 2D array graph, where graph[u] is an array of nodes that node u is adjacent to. More formally, for each v in graph[u], there is an undirected edge between node u and node v. The graph has the following properties:

• There are no self-edges (graph[u] does not contain u).
• There are no parallel edges (graph[u] does not contain duplicate values).
• If v is in graph[u], then u is in graph[v] (the graph is undirected).
• The graph may not be connected, meaning there may be two nodes u and v such that there is no path between them.

A graph is bipartite if the nodes can be partitioned into two independent sets A and B such that every edge in the graph connects a node in set A and a node in set B.

Return true if and only if it is bipartite.

Solution Approach

Expected Time complexity: $O(n)$

Click - to see solution code

C++

class Solution {
   public:
    bool dfs_helper(vector<vector<int>> Graph, vector<int>& visited, int par,
                    int node, int color) {
        visited[node] = color;          // painted
        for (auto nbr : Graph[node]) {  // traversing nbrs
            if (nbr != par and visited[nbr] == 0) {
                bool value = dfs_helper(Graph, visited, node, nbr, 3 - color);
                if (value == false) return false;
            } else if (visited[nbr] == color)
                return false;
        }
        return true;
    }

    bool dfs(vector<vector<int>> Graph, int N) {
        vector<int> visited(N + 1);
        int color = 1;
        for (int i = 0; i < N; i++) {
            if (visited[i] == 0) {
                bool ans = dfs_helper(Graph, visited, -1, i, 1);
                if (ans == false) return ans;
            }
        }
        return true;
    }

    bool isBipartite(vector<vector<int>>& graph) {
        int n = graph.size();
        return dfs(graph, n);
    }
};