# Minimum number of edges between two vertices of a graph using DFS

Given an undirected graph **G(V, E)** with **N** vertices and **M** edges. We need to find the minimum number of edges between a given pair of vertices **(u, v)**.

We have already discussed this problem using the BFS approach, here we will use the DFS approach.

**Examples:**

Input:For the following given graph, find the minimum number of edges between vertex pair (0, 4)

Output:1There are three paths from 0 to 4:

0 -> 1 -> 2 -> 4

0 -> 1 -> 2 -> 3 -> 4

0 -> 4

Only the third path results in minimum number of edges.

**Approach:** In this approach we will traverse the graph in a DFS manner, starting from the given vertex and explore all the paths from that vertex to our destination vertex.

We will use two variables, **edge_count** and **min_num_of_edges**. While exploring all the paths, between these vertices, **edge_count** will store count of total number of edges among them, if number of edges is less than the minimum number of edges we will update **min_num_of_edges**.

Below is the implementation of the above approach:

`// C++ program to find minimum ` `// number of edges between any two ` `// vertices of the graph ` ` ` `#include <bits/stdc++.h> ` `using` `namespace` `std; ` ` ` `// Class to represent a graph ` `class` `Graph { ` ` ` ` ` `// No. of vertices ` ` ` `int` `V; ` ` ` ` ` `// Pointer to an array containing ` ` ` `// adjacency lists ` ` ` `list<` `int` `>* adj; ` ` ` ` ` `// A function used by minEdgeDFS ` ` ` `void` `minEdgeDFSUtil(vector<` `bool` `>& visited, ` ` ` `int` `src, ` `int` `des, ` `int` `& min_num_of_edges, ` ` ` `int` `& edge_count); ` ` ` `public` `: ` ` ` `// Constructor ` ` ` `Graph(` `int` `V); ` ` ` ` ` `// Function to add an edge to graph ` ` ` `void` `addEdge(` `int` `src, ` `int` `des); ` ` ` ` ` `// Prints the minimum number of edges ` ` ` `void` `minEdgeDFS(` `int` `u, ` `int` `v); ` `}; ` ` ` `Graph::Graph(` `int` `V) ` `{ ` ` ` `this` `->V = V; ` ` ` `adj = ` `new` `list<` `int` `>[V]; ` `} ` ` ` `void` `Graph::addEdge(` `int` `src, ` `int` `des) ` `{ ` ` ` `adj[src].push_back(des); ` ` ` `adj[des].push_back(src); ` `} ` ` ` `// Utility function for finding minimum number ` `// of edges using DFS ` `void` `Graph::minEdgeDFSUtil(vector<` `bool` `>& visited, ` ` ` `int` `src, ` `int` `des, ` `int` `& min_num_of_edges, ` ` ` `int` `& edge_count) ` `{ ` ` ` `// For keeping track of visited ` ` ` `// nodes in DFS ` ` ` `visited[src] = ` `true` `; ` ` ` ` ` `// If we have found the destination vertex ` ` ` `// then check whether count of total number of edges ` ` ` `// is less than the minimum number of edges or not ` ` ` `if` `(src == des) { ` ` ` `if` `(min_num_of_edges > edge_count) ` ` ` `min_num_of_edges = edge_count; ` ` ` `} ` ` ` ` ` `// If current vertex is not destination ` ` ` `else` `{ ` ` ` ` ` `// Recur for all the vertices ` ` ` `// adjacent to current vertex ` ` ` `list<` `int` `>::iterator i; ` ` ` ` ` `for` `(i = adj[src].begin(); i != adj[src].end(); i++) { ` ` ` `int` `v = *i; ` ` ` ` ` `if` `(!visited[v]) { ` ` ` `edge_count++; ` ` ` ` ` `minEdgeDFSUtil(visited, v, des, min_num_of_edges, ` ` ` `edge_count); ` ` ` `} ` ` ` `} ` ` ` `} ` ` ` ` ` `// Decrement the count of number of edges ` ` ` `// and mark current vertex as unvisited ` ` ` `visited[src] = ` `false` `; ` ` ` `edge_count--; ` `} ` ` ` `// Function to print minimum number of edges ` `// It uses recursive minEdgeDFSUtil ` `void` `Graph::minEdgeDFS(` `int` `u, ` `int` `v) ` `{ ` ` ` `// To keep track of all the ` ` ` `// visited vertices ` ` ` `vector<` `bool` `> visited(V, ` `false` `); ` ` ` ` ` `// To store minimum number of edges ` ` ` `int` `min_num_of_edges = INT_MAX; ` ` ` ` ` `// To store total number of ` ` ` `// edges in each path ` ` ` `int` `edge_count = 0; ` ` ` ` ` `minEdgeDFSUtil(visited, u, v, min_num_of_edges, ` ` ` `edge_count); ` ` ` ` ` `// Print the minimum number of edges ` ` ` `cout << min_num_of_edges; ` `} ` ` ` `// Driver Code ` `int` `main() ` `{ ` ` ` `// Create a graph ` ` ` `Graph g(5); ` ` ` `g.addEdge(0, 1); ` ` ` `g.addEdge(0, 4); ` ` ` `g.addEdge(1, 2); ` ` ` `g.addEdge(2, 4); ` ` ` `g.addEdge(2, 3); ` ` ` `g.addEdge(3, 4); ` ` ` ` ` `int` `u = 0; ` ` ` `int` `v = 3; ` ` ` `g.minEdgeDFS(u, v); ` ` ` ` ` `return` `0; ` `} ` |

*chevron_right*

*filter_none*

**Output:**

2

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