What is K-connected Graph?

A k-connected graph is a type of graph where removing k-1 vertices (and edges) from the graph does not disconnect it. 

In other words, there are at least k distinct paths between any two vertices in the graph, and the graph remains connected even if k-1 vertices or edges are removed. The parameter k is known as the connectivity of the graph. The higher the connectivity of a graph, the more robust it is against failures or attacks.

A 4-connected Graph

Properties of k-connected Graph:

• Robustness: A k-connected graph is more robust against vertex or edge failures than a graph with lower connectivity. Removing k-1 vertices or edges from a k-connected graph does not disconnect it.
• Minimum degree: In a k-connected graph, every vertex has a degree of at least k. This means that the graph is densely connected and has many paths between its vertices.
• Separability: A graph is k-connected if and only if it is not separable into two disconnected subgraphs by removing k-1 vertices.
• Augmentation: Every k-connected graph can be augmented to a (k+1)-connected graph by adding edges between any two sets of k vertices that are not already connected.
• Planarity: In a planar k-connected graph, the number of vertices is at least 2k, and the number of edges is at least 3k – 6.

How to identify a k-connected Graph?

• Brute force approach: One way to identify if a graph is k-connected is to remove all possible subsets of k-1 vertices (or edges) and check if the graph remains connected after each removal. This approach is not efficient for large graphs, but it can be used for small graphs. 
• Max-flow min-cut theorem: According to the Max-flow min-cut theorem, the maximum flow in a graph is equal to the minimum cut in the graph. Therefore, we can use a max-flow algorithm to find the minimum cut of a graph, and if the minimum cut is greater than or equal to k, then the graph is k-connected.
• Connectivity algorithm: We can use a connectivity algorithm such as Tarjan’s algorithm or Hopcroft–Karp algorithm to find all the cut-vertices and cut-edges in a graph. If there are no cut-vertices and cut-edges of size less than k-1, then the graph is k-connected.
• DFS Algorithm: First we, check if K is greater than or equal to 2 and less than the total number of vertices in the graph. If K is greater than or equal to the number of vertices in the graph minus one, then the graph cannot be K-connected. Next, for each vertex in the graph, perform a DFS traversal and count the number of vertices that are visited. If the number of visited vertices is less than K, then the graph is not K-connected.
  If the above condition is not satisfied for any vertex in the graph, then the graph is K-connected.

Sudo code for dfs algorithm

function isKConnected(Graph G, int K):
    visited = {False} * |V(G)|
    if K < 2 or K >= |V(G)|:
        return False
    for each vertex u in G:
        count = 0
        DFS(u, visited, count)
        if count < K:
            return False
    return True
function DFS(vertex v, visited, count):
    visited[v] = True
    count += 1
    for each adjacent vertex w of v:
        if not visited[w]:
            DFS(w, visited, count)

Implementation:

C++

#include <bits/stdc++.h>
using namespace std;

const int N = 100005;

vector<int> adj[N];
bool visited[N];

void DFS(int u, int& count) {
    visited[u] = true;
    count++;
    for (int v : adj[u]) {
        if (!visited[v]) {
            DFS(v, count);
        }
    }
}

bool isKConnected(int n, int K) {
    if (K < 2 || K >= n) {
        return false;
    }
    for (int u = 0; u < n; u++) {
        memset(visited, false, sizeof(visited));
        int count = 0;
        DFS(u, count);
        if (count < K) {
            return false;
        }
    }
    return true;
}

int main() {
    int n, m, K;
      n = 5, m = 4, K = 2; // modify these values as needed
    adj[0] = {1, 2};
    adj[1] = {2, 3};
    adj[2] = {3, 4};
    adj[3] = {4, 5};
   /* cin >> n >> m >> K;
    for (int i = 0; i < m; i++) {
        int u, v;
        cin >> u >> v;
        adj[u].push_back(v);
        adj[v].push_back(u);
    }*/
    if (isKConnected(n, K)) {
        cout << "Graph is K-connected." << endl;
    } else {
        cout << "Graph is not K-connected." << endl;
    }
    return 0;
}

Java

import java.util.ArrayList;
import java.util.Arrays;
import java.util.List;

public class KConnectedGraph {

    static final int N = 100005;

    static List<Integer>[] adj = new ArrayList[N];
    static boolean[] visited = new boolean[N];

    static void DFS(int u, int[] count) {
        visited[u] = true;
        count[0]++;
        for (int v : adj[u]) {
            if (!visited[v]) {
                DFS(v, count);
            }
        }
    }

    static boolean isKConnected(int n, int K) {
        if (K < 2 || K >= n) {
            return false;
        }
        for (int u = 0; u < n; u++) {
            Arrays.fill(visited, false);
            int[] count = {0};
            DFS(u, count);
            if (count[0] < K) {
                return false;
            }
        }
        return true;
    }

    public static void main(String[] args) {
        int n = 5, m = 4, K = 2; // modify these values as needed

        for (int i = 0; i < N; i++) {
            adj[i] = new ArrayList<>();
        }

        adj[0].add(1);
        adj[0].add(2);
        adj[1].add(2);
        adj[1].add(3);
        adj[2].add(3);
        adj[2].add(4);
        adj[3].add(4);
        adj[3].add(5);

        /* Uncomment the following section to take input from the user
        Scanner scanner = new Scanner(System.in);
        n = scanner.nextInt();
        m = scanner.nextInt();
        K = scanner.nextInt();

        for (int i = 0; i < m; i++) {
            int u = scanner.nextInt();
            int v = scanner.nextInt();
            adj[u].add(v);
            adj[v].add(u);
        }*/

        if (isKConnected(n, K)) {
            System.out.println("Graph is K-connected.");
        } else {
            System.out.println("Graph is not K-connected.");
        }
    }
}

Python3

# Set recursion limit to prevent RecursionError for large graphs
import sys
sys.setrecursionlimit(10**6)

# Maximum number of nodes in the graph
N = 100005

# Create an adjacency list to store the graph
adj = [[] for _ in range(N)]

# Boolean array to keep track of visited nodes
visited = [False] * N

# Depth-first search function to visit all nodes in a connected component
def DFS(u, count):
    visited[u] = True
    count[0] += 1
    for v in adj[u]:
        if not visited[v]:
            DFS(v, count)

# Function to check if the graph is K-connected
def isKConnected(n, K):
    # If K is less than 2 or greater than or equal to n, then the graph is not K-connected
    if K < 2 or K >= n:
        return False
    # For each node u in the graph, perform a DFS to count the number of nodes reachable from u
    for u in range(n):
        global visited
        visited = [False] * n
        count = [0]
        DFS(u, count)
        # If the number of reachable nodes is less than K, then the graph is not K-connected
        if count[0] < K:
            return False
    # If all nodes satisfy the condition, then the graph is K-connected
    return True

# Example input values
n = 5
m = 4
K = 2
edges = [(1, 2), (2, 3), (3, 4), (4, 5)]

# Add input edges to the adjacency list
for u, v in edges:
    adj[u].append(v)
    adj[v].append(u)

# Check if the graph is K-connected and print the result
if isKConnected(n, K):
    print("Graph is K-connected.")
else:
    print("Graph is not K-connected.")
#Contributed by siddharth aher

C#

using System;
using System.Collections.Generic;

class Program {
    const int N = 100005;

    static List<int>[] adj = new List<int>[ N ];
    static bool[] visited = new bool[N];

    static void DFS(int u, ref int count)
    {
        visited[u] = true;
        count++;
        foreach(int v in adj[u])
        {
            if (!visited[v]) {
                DFS(v, ref count);
            }
        }
    }

    static bool IsKConnected(int n, int K)
    {
        if (K < 2 || K >= n) {
            return false;
        }

        for (int u = 0; u < n; u++) {
            Array.Fill(visited, false);
            int count = 0;
            DFS(u, ref count);
            if (count < K) {
                return false;
            }
        }
        return true;
    }

    static void Main()
    {
        int n, K;
        n = 5;
        K = 2; // modify these values as needed

        // Initialize adjacency list
        for (int i = 0; i < N; i++) {
            adj[i] = new List<int>();
        }

        adj[0].AddRange(new[] { 1, 2 });
        adj[1].AddRange(new[] { 2, 3 });
        adj[2].AddRange(new[] { 3, 4 });
        adj[3].AddRange(new[] { 4, 5 });

        /* Uncomment the following block to take input from
        the user Console.WriteLine("Enter the number of
        vertices, edges, and K:"); string[] input =
        Console.ReadLine().Split(); n = int.Parse(input[0]);
        K = int.Parse(input[1]);

        Console.WriteLine("Enter the edges:");
        for (int i = 0; i < m; i++)
        {
            int u, v;
            input = Console.ReadLine().Split();
            u = int.Parse(input[0]);
            v = int.Parse(input[1]);
            adj[u].Add(v);
            adj[v].Add(u);
        }
        */

        if (IsKConnected(n, K)) {
            Console.WriteLine("Graph is K-connected.");
        }
        else {
            Console.WriteLine("Graph is not K-connected.");
        }
    }
}

Javascript

const N = 100005;
const adj = Array.from({ length: N }, () => []);
const visited = Array(N).fill(false);

function DFS(u, count) {
    visited[u] = true;
    count[0]++;
    for (const v of adj[u]) {
        if (!visited[v]) {
            DFS(v, count);
        }
    }
}

function isKConnected(n, K) {
    if (K < 2 || K >= n) {
        return false;
    }
    for (let u = 0; u < n; u++) {
        visited.fill(false);
        const count = [0];
        DFS(u, count);
        if (count[0] < K) {
            return false;
        }
    }
    return true;
}

// Example usage:
const n = 5, m = 4, K = 2; // Modify these values as needed
adj[0] = [1, 2];
adj[1] = [2, 3];
adj[2] = [3, 4];
adj[3] = [4, 5];

if (isKConnected(n, K)) {
    console.log("Graph is K-connected.");
} else {
    console.log("Graph is not K-connected.");
}

Output

Graph is not K-connected.