using dfs to check if graph is bipartite

Question

class graph{
    void DFSutil(int v);

    public:
        
        map<int, vector<int> > adj;
        map<int, bool> visited;
        map<int, int> color;
        int twoteams = 1;
        
        void DFS();
        void addEdge(int u,int v);
        
        
};

this function adds edge to the graph

void graph::addEdge(int u,int v){
    
    adj[u].push_back(v);
    
}

depth first search function. i tried to color the child as !parent (in case im not wrong): and if the color of child and parent is same, that means the graph is not bipartite

void graph::DFSutil(int v){
    
    visited[v] = true;
    
    for (auto i:adj[v]){
        if(visited[i]==false){
            
            color[i] = !color[v];
            DFSutil(i);
            
        }
        else if(color[i]==color[v]){
            
        twoteams = 0;
        return;
        }
    }
}

in case the graph has more than one connected component

void graph::DFS(){
    
    for (auto i:adj){
        if(!visited[i.first]) DFSutil(i.first);
    }
    
}

the graph is given as: B (a vector): [[1,2],[1,3]...] the edges: 1-->2, 1-->3...

int Solution::solve(int A, vector<vector<int> > &B) {
    
    graph g;
    g.addEdge(B[0][0],B[0][1]);
    g.color[B[0][0]] = 1;
    int n = B.size();
    for (int i=1;i<n;++i){
        
        g.addEdge(B[i][0],B[i][1]);
        
    }
    g.DFS();
    return g.twoteams;
}

why does it not check if the graph is bipartite. color stores the color of a node as 0 or 1; it should return 0 if not bipartite and 1 otherwise

Answer 1

To determine if a graph is bipartite it is sufficient to determine that the number of maximal cliques is exactly two.

Here is the pseudo code for finding the maximal cliques

LOOP
    CONSTRUCT empty current set
    SELECT V arbitrary vertex
    add V to current set
    remove V from graph
    LOOP      // while set is growing
        added_to_set = false
        LOOP V over vertices in graph
            LOOP Vset over current set
                IF Vset connected to V
                     add V to current set
                     remove V from graph
                     added_to_set = true
                     break;
        IF added_to_set == false
           break;    // the set is maximal
    ADD current set to list of sets
    IF graph has no remaining vertices
       OUTPUT sets found
       STOP

For a C++ implementation of this see code at https://github.com/JamesBremner/PathFinder2/blob/dbd6ff06edabd6a6d35d5eb10ed7972dc2d779a6/src/cPathFinder.cpp#L483