Concepts Used

Depth First Search, Graph

Difficulty Level

Medium

Problem Statement :

Check whether the graph is a tree or not.

See original problem statement here

Solution Approach :

Introduction :

For a graph to be tree, there should not be any loops and every vertex must be reachable from atleast one other vertex.

This problem can be solved by many ways like Breadth First Search, Depth First Search or Disjoint Set. Below we are going to go through some algorithms for beginners and discuss two of the above mentioned methods to solve this problem.

Method 1 :

The graph must follow these properties:

  • If there are n vertices then there must be n-1 edges.
  • It should be connected i.e. every vertex can be reached with atleast one other vertex.

In trees, every node/vertex is connected to atleast one other vertex. Also the total number of edges is also n-1 for n nodes.

Method 2 :

The another way of checking our graph whether it is a tree or not that it should have following properties:

  • It must not contain any cycle.
  • It must be connected.

As, we can see this approach is almost similar to the approach in method 1, our goal is to make sure that our graph has no loops or in other words it has exactly n-1 edges and must be connected. By this way we can ensure our graph is a tree, otherwise not. Refer to the example below for better understanding.

See C++ implementation below.

Algorithms :

dfs() :

  1. For each call, for some vertex v ( dfs(v) ), we will mark the vertex v as visited (visited[v]= true).
  2. Iterate for all the adjacent vertices of v and for every adjacent vertex a, do following :
    • if a is not visited i.e visited[a]= false,
      and if a has value 1.
      recursively call dfs (a).

Complexity Analysis:

The time complexity of the first method is represented in the form of O(V+E), where V is the number of verices and E is the number of edges.

The space complexity of the algorithm is O(V) for visited[] array.

Solutions:


#include <stdio.h>
    #include <stdlib.h>
    #include<string.h>
    #define INT_MIN -99999

    //ADJACENCY LIST
    struct node {
      int vertex;
      struct node* next;
    };
    struct node* createNode(int);

    struct Graph {
      int numVertices;
      struct node** adjLists;
    };

    // Create a node
    struct node* createNode(int v) {
      struct node* newNode = malloc(sizeof(struct node));
      newNode->vertex = v;
      newNode->next = NULL;
      return newNode;
    }

    // Create a graph
    struct Graph* createAGraph(int vertices) {
      struct Graph* graph = malloc(sizeof(struct Graph));
      graph->numVertices = vertices;

      graph->adjLists = malloc(vertices * sizeof(struct node*));

      int i;
      for (i = 0; i < vertices; i++)
        graph->adjLists[i] = NULL;

      return graph;
    }

    // Add edge
    void addEdge(struct Graph* graph, int s, int d) {
      // Add edge from s to d
      struct node* newNode = createNode(d);
      newNode->next = graph->adjLists[s];
      graph->adjLists[s] = newNode;

      // Add edge from d to s
      newNode = createNode(s);
      newNode->next = graph->adjLists[d];
      graph->adjLists[d] = newNode;
    }


     void dfs(struct Graph *adj,int visited[], int v)
    {
        visited[v]= 1;

        struct node *u = adj->adjLists[v]; 
        while(u)
        //for (i = adj[v].begin(); i != adj[v].end(); ++i)
        {
        if(!visited[u->vertex])
         dfs(adj,visited,u->vertex);
         u= u->next;
        }
    }


       int isTree(struct Graph *adj,int n) 
    { 

        int visited[n]; 
        for (int i = 0; i < n; i++) 
            {visited[i] = 0; }


            dfs(adj,visited,0);

        for (int u = 0; u < n; u++) 
            if (!visited[u]) 
            return 0; 

        return 1; 
      } 

    int main() 
    { 
    int t;
    scanf("%d",&t);
    while(t--){
    int n,e;
    scanf("%d %d",&n,&e);

     struct Graph* graph = createAGraph(n);
       int edge = e;
        while(e--)
        {
          int u,v;
          scanf("%d %d",&u,&v);
          addEdge(graph, u,v);

         }
             if(edge == n-1 && isTree(graph,n))
              printf("YES\n");
             else
              printf("NO\n");
        }

        return 0; 
    } 

#include<iostream> 
    #include <list> 
    #include <limits.h> 
    using namespace std; 


    class Graph 
    { 
    int V; 
    list<int> *adj; 
    bool isCyclicUtil(int v, bool visited[], int parent); 
    public: 
    Graph(int V); // Constructor 
    void addEdge(int v, int w); // to add an edge to graph 
    void dfs(bool visited[],int i);
    bool isTree(); // returns true if graph is tree 
    }; 

    Graph::Graph(int V) 
    { 
    this->V = V; 
    adj = new list<int>[V]; 
    } 

    void Graph::dfs(bool visited[], int v)
    {
        visited[v]= true;

        list<int>::iterator i; 
        for (i = adj[v].begin(); i != adj[v].end(); ++i)
        {
        if(!visited[*i])
         dfs(visited,*i);
        }
    }

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


    bool Graph::isTree() 
    { 

    bool *visited = new bool[V]; 
    for (int i = 0; i < V; i++) 
        visited[i] = false; 


        dfs(visited,0);

    for (int u = 0; u < V; u++) 
        if (!visited[u]) 
        return false; 

    return true; 
    } 

    int main() 
    { 
    int t;
    cin>>t;
    while(t--){
    int n,e;
    cin>>n>>e;

    Graph g1(n); 
    int edge = e;
    while(e--)
    {
      int u,v;
      cin>>u>>v;
      g1.addEdge(u,v); 

     }
         if(edge == n-1 && g1.isTree())
          cout<<"YES"<<endl;
         else
          cout<<"NO"<<endl;
    }

    return 0; 
    } 

import java.io.*; 
    import java.util.*; 


    class Graph 
    { 
        private int V; // No. of vertices 
        private LinkedList<Integer> adj[]; //Adjacency List 

    // Constructor 
    Graph(int v) 
    { 
        V = v; 
        adj = new LinkedList[v]; 
        for (int i=0; i<v; ++i) 
            adj[i] = new LinkedList(); 
    } 

    // Function to add an edge into the graph 
    void addEdge(int v,int w) 
    { 
        adj[v].add(w); 
        adj[w].add(v); 
    } 

    Boolean isCyclicUtil(int v, Boolean visited[], int parent) 
    { 
        visited[v] = true; 
        Integer i; 

        Iterator<Integer> it = adj[v].iterator(); 
        while (it.hasNext()) 
        { 
            i = it.next(); 

            if (!visited[i]) 
            { 
                if (isCyclicUtil(i, visited, v)) 
                    return true; 
            } 

            else if (i != parent) 
            return true; 
        } 
        return false; 
    } 

    Boolean isTree() 
    { 
        Boolean visited[] = new Boolean[V]; 
        for (int i = 0; i < V; i++) 
            visited[i] = false; 

        if (isCyclicUtil(0, visited, -1)) 
            return false; 

        for (int u = 0; u < V; u++) 
            if (!visited[u]) 
                return false; 

        return true; 
    } 

    public static void main(String args[]) 
    { 
      Scanner sc = new Scanner(System.in);
      int t= sc.nextInt();
      while(t-- !=0)
      {
        int n,e;
        n = sc.nextInt();
        e = sc.nextInt();

        Graph g1 = new Graph(n); 
        while(e--!=0)
        {
          int u,v;
          u = sc.nextInt();
          v = sc.nextInt();
        g1.addEdge(u,v); 

        }
        if (g1.isTree()) 
            System.out.println("YES"); 
        else
            System.out.println("NO"); 
      } 
     } 
    } 

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