An undirected graph is called Biconnected if there are two vertex-disjoint paths between any two vertices. In a Biconnected Graph, there is a simple cycle through any two vertices.
By convention, two nodes connected by an edge form a biconnected graph, but this does not verify the above properties. For a graph with more than two vertices, the above properties must be there for it to be Biconnected.
Following are some examples.
How to find if a given graph is Biconnected or not?
A connected graph is Biconnected if it is connected and doesn’t have any Articulation Point. We mainly need to check two things in a graph.
1) The graph is connected.
2) There is not articulation point in graph.
We start from any vertex and do DFS traversal. In DFS traversal, we check if there is any articulation point. If we don’t find any articulation point, then the graph is Biconnected. Finally, we need to check whether all vertices were reachable in DFS or not. If all vertices were not reachable, then the graph is not even connected.
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Java programming:
import java.io.*;
import java.util.*;
import java.util.LinkedList;
class Graph
{
private int V;
private LinkedList<Integer> adj[];
int time = 0;
static final int NIL = -1;
Graph(int v)
{
V = v;
adj = new LinkedList[v];
for (int i=0; i<v; ++i)
adj[i] = new LinkedList();
}
void addEdge(int v, int w)
{
adj[v].add(w);
adj[w].add(v);
}
boolean isBCUtil(int u, boolean visited[], int disc[],int low[],
int parent[])
{
int children = 0;
visited[u] = true;
disc[u] = low[u] = ++time;
Iterator<Integer> i = adj[u].iterator();
while (i.hasNext())
{
int v = i.next();
if (!visited[v])
{
children++;
parent[v] = u;
if (isBCUtil(v, visited, disc, low, parent))
return true;
low[u] = Math.min(low[u], low[v]);
if (parent[u] == NIL && children > 1)
return true;
if (parent[u] != NIL && low[v] >= disc[u])
return true;
}
else if (v != parent[u])
low[u] = Math.min(low[u], disc[v]);
}
return false;
}
boolean isBC()
{
boolean visited[] = new boolean[V];
int disc[] = new int[V];
int low[] = new int[V];
int parent[] = new int[V];
for (int i = 0; i < V; i++)
{
parent[i] = NIL;
visited[i] = false;
}
if (isBCUtil(0, visited, disc, low, parent) == true)
return false;
for (int i = 0; i < V; i++)
if (visited[i] == false)
return false;
return true;
}
public static void main(String args[])
{
Graph g1 =new Graph(2);
g1.addEdge(0, 1);
if (g1.isBC())
System.out.println("Yes");
else
System.out.println("No");
Graph g2 =new Graph(5);
g2.addEdge(1, 0);
g2.addEdge(0, 2);
g2.addEdge(2, 1);
g2.addEdge(0, 3);
g2.addEdge(3, 4);
g2.addEdge(2, 4);
if (g2.isBC())
System.out.println("Yes");
else
System.out.println("No");
Graph g3 = new Graph(3);
g3.addEdge(0, 1);
g3.addEdge(1, 2);
if (g3.isBC())
System.out.println("Yes");
else
System.out.println("No");
Graph g4 = new Graph(5);
g4.addEdge(1, 0);
g4.addEdge(0, 2);
g4.addEdge(2, 1);
g4.addEdge(0, 3);
g4.addEdge(3, 4);
if (g4.isBC())
System.out.println("Yes");
else
System.out.println("No");
Graph g5= new Graph(3);
g5.addEdge(0, 1);
g5.addEdge(1, 2);
g5.addEdge(2, 0);
if (g5.isBC())
System.out.println("Yes");
else
System.out.println("No");
}
}
Output:
Yes
Yes
No
No
Yes
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