Check if a given graph is tree or not

// A C++ Program to check whether a graph is tree or not
#include<iostream>
#include <list>
#include <limits.h>
using namespace std;

// Class for an undirected graph
class Graph
{
    int V;    // No. of vertices
    list<int> *adj; // Pointer to an array for adjacency lists
    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
    bool isTree();   // returns true if graph is tree
};

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

void Graph::addEdge(int v, int w)
{
    adj[v].push_back(w); // Add w to v’s list.
    adj[w].push_back(v); // Add v to w’s list.
}

// A recursive function that uses visited[] and parent to
// detect cycle in subgraph reachable from vertex v.
bool Graph::isCyclicUtil(int v, bool visited[], int parent)
{
    // Mark the current node as visited
    visited[v] = true;

    // Recur for all the vertices adjacent to this vertex
    list<int>::iterator i;
    for (i = adj[v].begin(); i != adj[v].end(); ++i)
    {
        // If an adjacent is not visited, then recur for 
        // that adjacent
        if (!visited[*i])
        {
           if (isCyclicUtil(*i, visited, v))
              return true;
        }

        // If an adjacent is visited and not parent of current
        // vertex, then there is a cycle.
        else if (*i != parent)
           return true;
    }
    return false;
}

// Returns true if the graph is a tree, else false.
bool Graph::isTree()
{
    // Mark all the vertices as not visited and not part of 
    // recursion stack
    bool *visited = new bool[V];
    for (int i = 0; i < V; i++)
        visited[i] = false;

    // The call to isCyclicUtil serves multiple purposes.
    // It returns true if graph reachable from vertex 0 
    // is cyclic. It also marks all vertices reachable 
    // from 0.
    if (isCyclicUtil(0, visited, -1))
             return false;

    // If we find a vertex which is not reachable from 0 
    // (not marked by isCyclicUtil(), then we return false
    for (int u = 0; u < V; u++)
        if (!visited[u])
           return false;

    return true;
}

// Driver program to test above functions
int main()
{
    Graph g1(5);
    g1.addEdge(1, 0);
    g1.addEdge(0, 2);
    g1.addEdge(0, 3);
    g1.addEdge(3, 4);
    g1.isTree()? cout << "Graph is Tree\n":
                 cout << "Graph is not Tree\n";

    Graph g2(5);
    g2.addEdge(1, 0);
    g2.addEdge(0, 2);
    g2.addEdge(2, 1);
    g2.addEdge(0, 3);
    g2.addEdge(3, 4);
    g2.isTree()? cout << "Graph is Tree\n":
                 cout << "Graph is not Tree\n";

    return 0;
}

// A C++ Program to check whether a graph is tree or not
#include<iostream>
#include <list>
#include <limits.h>
using namespace std;

// Class for an undirected graph
class Graph
{
    int V;    // No. of vertices
      int E;    // No. of edges
    list<int> *adj; // Pointer to an array for adjacency lists
    void dfsTraversal(int v, bool visited[], int parent);
public:
    Graph(int V);   // Constructor
    void addEdge(int v, int w);   // to add an edge to graph
    bool isConnected();   // returns true if graph is connected
    bool isTree();     // returns true of the graph is tree
};

Graph::Graph(int V)
{
    E = 0;
    this->V = V;
    adj = new list<int>[V];
}

void Graph::addEdge(int v, int w)
{
    E++;                 // increase the number of edges
    adj[v].push_back(w); // Add w to v’s list.
    adj[w].push_back(v); // Add v to w’s list.
}

// A recursive dfs function that uses visited[] and parent to
// traverse the graph and mark visited[v] to true for visited nodes
void Graph::dfsTraversal(int v, bool visited[], int parent)
{
    // Mark the current node as visited
    visited[v] = true;

    // Recur for all the vertices adjacent to this vertex
    list<int>::iterator i;
    for (i = adj[v].begin(); i != adj[v].end(); ++i)
    {
        // If an adjacent is not visited, then recur for 
        // that adjacent
        if (!visited[*i])
        {
           dfsTraversal(*i, visited, v);
        }
    }
}

// Returns true if the graph is connected, else false.
bool Graph::isConnected()
{
    // Mark all the vertices as not visited and not part of 
    // recursion stack
    bool *visited = new bool[V];
    for (int i = 0; i < V; i++)
        visited[i] = false;

    // Performing DFS traversal of the graph and marking
    // reachable vertices from 0 to true
    dfsTraversal(0, visited, -1);

    // If we find a vertex which is not reachable from 0 
    // (not marked by dfsTraversal(), then we return false
    // since graph is not connected
    for (int u = 0; u < V; u++)
        if (!visited[u])
           return false;

    // since all nodes were reachable so we returned true and
    // and hence graph is connected
    return true;
}

bool Graph::isTree()
{
    // as we proved earlier if a graph is connected and has
    // V - 1 edges then it is a tree i.e. E = V - 1
    return isConnected() and E == V - 1;
}
// Driver program to test above functions
int main()
{
    Graph g1(5);
    g1.addEdge(1, 0);
    g1.addEdge(0, 2);
    g1.addEdge(0, 3);
    g1.addEdge(3, 4);
    g1.isTree()? cout << "Graph is Tree\n":
                 cout << "Graph is not Tree\n";

    Graph g2(5);
    g2.addEdge(1, 0);
    g2.addEdge(0, 2);
    g2.addEdge(2, 1);
    g2.addEdge(0, 3);
    g2.addEdge(3, 4);
    g2.isTree()? cout << "Graph is Tree\n":
                 cout << "Graph is not Tree\n";

    return 0;
}