# Check if removing a given edge disconnects a graph

Given an undirected graph and an edge, the task is to find if the given edge is a bridge in graph, i.e., removing the edge disconnects the graph.

Following are some example graphs with bridges highlighted with red color.

One solution is to find all bridges in given graph and then check if given edge is a bridge or not.

A simpler solution is to remove the edge, check if graph remains connect after removal or not, finally add the edge back. We can always find if an undirected is connected or not by finding all reachable vertices from any vertex. If count of reachable vertices is equal to number of vertices in graph, then the graph is connected else not. We can find all reachable vertices either using BFS or DFS. Below are complete steps.

1) Remove the given edge
2) Find all reachable vertices from any vertex. We have chosen first vertex in below implementation.
3) If count of reachable nodes is V, then return false [given is not Bridge]. Else return yes.

## C++

 `// C++ program to check if removing an ` `// edge disconnects a graph or not. ` `#include ` `using` `namespace` `std; ` ` `  `// Graph class represents a directed graph ` `// using adjacency list representation ` `class` `Graph ` `{ ` `    ``int` `V;    ``// No. of vertices ` `    ``list<``int``> *adj; ` `    ``void` `DFS(``int` `v, ``bool` `visited[]); ` `public``: ` `    ``Graph(``int` `V);   ``// Constructor ` ` `  `    ``// function to add an edge to graph ` `    ``void` `addEdge(``int` `v, ``int` `w); ` ` `  `    ``// Returns true if graph is connected ` `    ``bool` `isConnected(); ` ` `  `    ``bool` `isBridge(``int` `u, ``int` `v); ` `}; ` ` `  `Graph::Graph(``int` `V) ` `{ ` `    ``this``->V = V; ` `    ``adj = ``new` `list<``int``>[V]; ` `} ` ` `  `void` `Graph::addEdge(``int` `u, ``int` `v) ` `{ ` `    ``adj[u].push_back(v); ``// Add w to v’s list. ` `    ``adj[v].push_back(u); ``// Add w to v’s list. ` `} ` ` `  `void` `Graph::DFS(``int` `v, ``bool` `visited[]) ` `{ ` `    ``// Mark the current node as visited and print it ` `    ``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` `(!visited[*i]) ` `            ``DFS(*i, visited); ` `} ` ` `  `// Returns true if given graph is connected, else false ` `bool` `Graph::isConnected() ` `{ ` `    ``bool` `visited[V]; ` `    ``memset``(visited, ``false``, ``sizeof``(visited)); ` ` `  `    ``// Find all reachable vertices from first vertex ` `    ``DFS(0, visited); ` ` `  `    ``// If set of reachable vertices includes all, ` `    ``// return true. ` `    ``for` `(``int` `i=1; i

## Python3

 `# Python3 program to check if removing  ` `# an edge disconnects a graph or not.  ` ` `  `# Graph class represents a directed graph  ` `# using adjacency list representation  ` `class` `Graph: ` ` `  `    ``def` `__init__(``self``, V): ` `        ``self``.V ``=` `V  ` `        ``self``.adj ``=` `[[] ``for` `i ``in` `range``(V)] ` `     `  `    ``def` `addEdge(``self``, u, v): ` `        ``self``.adj[u].append(v) ``# Add w to v’s list.  ` `        ``self``.adj[v].append(u) ``# Add w to v’s list. ` `     `  `    ``def` `DFS(``self``, v, visited): ` `         `  `        ``# Mark the current node as  ` `        ``# visited and print it  ` `        ``visited[v] ``=` `True` `     `  `        ``# Recur for all the vertices  ` `        ``# adjacent to this vertex ` `        ``i ``=` `0` `        ``while` `i !``=` `len``(``self``.adj[v]): ` `            ``if` `(``not` `visited[``self``.adj[v][i]]):  ` `                ``self``.DFS(``self``.adj[v][i], visited) ` `            ``i ``+``=` `1` `     `  `    ``# Returns true if given graph is  ` `    ``# connected, else false  ` `    ``def` `isConnected(``self``): ` `        ``visited ``=` `[``False``] ``*` `self``.V ` `     `  `        ``# Find all reachable vertices  ` `        ``# from first vertex  ` `        ``self``.DFS(``0``, visited)  ` `     `  `        ``# If set of reachable vertices  ` `        ``# includes all, return true. ` `        ``for` `i ``in` `range``(``1``, ``self``.V): ` `            ``if` `(visited[i] ``=``=` `False``):  ` `                ``return` `False` `     `  `        ``return` `True` ` `  `    ``# This function assumes that edge   ` `    ``# (u, v) exists in graph or not,  ` `    ``def` `isBridge(``self``, u, v): ` `         `  `        ``# Remove edge from undirected graph ` `        ``indU ``=` `self``.adj[v].index(u) ` `        ``indV ``=` `self``.adj[u].index(v) ` `        ``del` `self``.adj[u][indV]  ` `        ``del` `self``.adj[v][indU]  ` `     `  `        ``res ``=` `self``.isConnected()  ` `     `  `        ``# Adding the edge back  ` `        ``self``.addEdge(u, v)  ` `     `  `        ``# Return true if graph becomes  ` `        ``# disconnected after removing  ` `        ``# the edge.  ` `        ``return` `(res ``=``=` `False``) ` ` `  `# Driver code  ` `if` `__name__ ``=``=` `'__main__'``: ` ` `  `    ``# Create a graph given in the ` `    ``# above diagram  ` `    ``g ``=` `Graph(``4``) ` `    ``g.addEdge(``0``, ``1``)  ` `    ``g.addEdge(``1``, ``2``)  ` `    ``g.addEdge(``2``, ``3``)  ` ` `  `    ``if` `g.isBridge(``1``, ``2``): ` `        ``print``(``"Yes"``) ` `    ``else``: ` `        ``print``(``"No"``) ` ` `  `# This code is contributed by PranchalK `

Output :

`Yes`

Time Complexity : O(V + E)

Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above

This article is attributed to GeeksforGeeks.org

## tags:

Graph graph-connectivity Graph

code

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