This algorithm finds the connected components of an undirected graph using depth-first search (DFS) traversal. The resulting vector of components is then printed out, along with the number of components found.

Time Complexity: O(m+n), where m is the number of edges and n is the number of vertices

           
            vector<bool> visited;
            vector<int> finish_times;
            vector<vector<int>> components = {};
            
            void find_connected_components(vector<vector<int>> &graph)
            {
                visited.resize(graph.size(), false);
            
                for (int i = 0; i < graph.size(); i++)
                {
                    if (!visited.at(i))
                    {
                        components.push_back({});
                        DFS(graph, i);
                    }
                }
            }
            
            void DFS(vector<vector<int>> &graph, int vertix)
            {
            
                if (!visited.empty())
                {
                    visited.resize(graph.size(), false);
                }
            
                visited.at(vertix) = true;
                components.at(components.size() - 1).push_back(vertix);
            
                for (int i = 0; i < graph.at(vertix).size(); i++)
                {
            
                    if (!visited.at(graph.at(vertix).at(i)))
                    {
                        DFS(graph, graph.at(vertix).at(i));
                    }
                }
            }
            
            // Example Run
            int main()
            {
            
                //  Sample Adjecency Listx
                vector<vector<int>> graph = {
                    {1, 2, 3},
                    {0, 2, 3},
                    {0, 1, 3},
                    {0, 1, 2},
                    {5},
                    {4},
                    {7, 8},
                    {6, 8},
                    {6, 7},
                };
                // Sample Run
            
                find_connected_components(graph);
            
                cout << "Number of Components: " << components.size() << endl;
            
                for (int i = 0; i < components.size(); i++)
                {
                    cout << "Component " << i + 1 << ": " << endl;
                    for (int j = 0; j < components.at(i).size(); j++)
                    {
                        cout << components.at(i).at(j) << " ";
                    }
                    cout << endl;
                }
            }