Find All Forward Edges in a Graph using C++

In this article, we will learn how to write an algorithm an C++ code to find all forward edges in a directed graph.

What is a Forward Edge?

A forward edge is an edge in a directed graph that points from a node to one of it's descendants in the depth-first search (DFS) tree. To understand this concept better, consider the image of a directed graph below:

In the above graph, the edge from node 2 to node 5 is a forward edge because to reach node 5 in DFS traversal, we need to move through node 2 -> node 4 -> node 5. In other words, forward edges in a graph is an edge that bypasses the actual path in the DFS tree.

Find Forward Edges in a Graph

We can slightly modify the DFS algorithm to find forward edges in a directed graph. We will keep track of the time when a node was first visited and the time when we finish DFS for the node. We will use two arrays, in_time and out_time, to store these times for each node.

• in-time: The time when a node is first visited during DFS.
• out-time: The time when we finish DFS for the node.

Then for any edge (u -> v), if in_time[u] < in_time[v] and out_time[u] > out_time[v], then (u -> v) is a forward edge.

Algorithm to Find Forward Edges in a Graph

Here is the algorithm to find all forward edges in a directed graph:

• Initialize the graph using an adjacency list.
• Initialize arrays in_time and out_time to store the in-time and out-time of each node.
• Now, perform a DFS traversal of the graph using a recursive function. Store in-time while entering the function and out-time when exiting the function.
• For each edge (u -> v), check if it is a forward edge using the conditions mentioned above.
• Store and print all forward edges found.

C++ Code to Find Forward Edges in a Graph

Here is a C++ code that implements the above algorithm to find all forward edges in a directed graph:

#include <iostream>
#include <vector>
using namespace std;

vector<vector<int>> adj;
vector<bool> visited;
vector<int> inTime, outTime;
int timer = 0;

// DFS Function
void dfs(int u) {
    visited[u] = true;
    inTime[u] = ++timer;

    for (int v : adj[u]) {
        if (!visited[v]) {
            dfs(v);  // Tree Edge
        } else {
            // Check if it's a forward edge
            if (inTime[u] < inTime[v] && outTime[v] != 0) {
                cout << "Forward Edge: " << u << " -> " << v << endl;
            }
        }
    }

    outTime[u] = ++timer;
}

int main() {
    int V = 6;
    adj.resize(V);
    visited.resize(V, false);
    inTime.resize(V, 0);
    outTime.resize(V, 0);

    // Directed graph edges
    adj[0].push_back(1);
    adj[0].push_back(2);
    adj[1].push_back(3);
    adj[2].push_back(3);
    adj[3].push_back(4);
    adj[1].push_back(5);
    adj[0].push_back(4); // forward edge expected here

    for (int i = 0; i < V; ++i) {
        if (!visited[i])
            dfs(i);
    }

    return 0;
}

Output of the above code will be:

Forward Edge: 0 -> 4

Time and Space Complexity

Time Complexity: The time complexity of this algorithm is O(V + E), where V is the number of vertices and E is the number of edges in the graph. This is the same as the time complexity of DFS.

Space Complexity: The space complexity is also O(V) for storing the adjacency list and the visited, inTime, and outTime arrays.