Check if every vertex triplet in graph contains two vertices connected to third vertex

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Given an undirected graph with N vertices and K edges, the task is to check if for every combination of three vertices in the graph, there exists two vertices which are connected to third vertex. In other words, for every vertex triplet (a, b, c), if there exists a path between a and c, then there should also exist a path between b and c.

Examples:

Input: N = 4, K = 3
Edges: 1 -> 2, 2 -> 3, 3 -> 4
Output: YES
Explanation:
Since the whole graph is connected, the above condition will always be valid.

Input: N = 5 and K = 3
Edges: 1 -> 3, 3 -> 4, 2 -> 5.
Output: NO
Explanation:
If we consider the triplet (1, 2, 3) then there is a path between vertices 1 and 3 but there is no path between vertices 2 and 3.

Approach: Follow the steps below to solve the problem –

Traverse the graph by DFS Traversal technique from any component and maintain two variables to store the component minimum and component maximum.
Store every component maximum and minimum in a vector.
Now, if any two components have an intersection in their minimum and maximum values interval, then there will exist a valid (a < b < c) triplet. Hence, both of the components should be connected. Otherwise, the graph is not valid.

Below is the implementation of the above approach

C++

// C++ program of the
// above approach
 
#include <bits/stdc++.h>
using namespace std;
 
// Function to add edge into
// the graph
void addEdge(vector<int> adj[],
             int u, int v)
{
    adj[u].push_back(v);
    adj[v].push_back(u);
}
 
void DFSUtil(int u, vector<int> adj[],
             vector<bool>& visited,
             int& componentMin,
             int& componentMax)
{
    visited[u] = true;
 
    // Finding the maximum and
    // minimum values in each component
    componentMax = max(componentMax, u);
    componentMin = min(componentMin, u);
 
    for (int i = 0; i < adj[u].size(); i++)
        if (visited[adj[u][i]] == false)
            DFSUtil(adj[u][i], adj, visited,
                    componentMin, componentMax);
}
 
// Function for checking whether
// the given graph is valid or not
bool isValid(vector<pair<int, int> >& v)
{
    int MAX = -1;
    bool ans = 0;
    // Checking for intersecting intervals
    for (auto i : v) {
        // If intersection is found
        if (i.first <= MAX) {
 
            // Graph is not valid
            ans = 1;
        }
 
        MAX = max(MAX, i.second);
    }
 
    return (ans == 0 ? 1 : 0);
}
 
// Function for the DFS Traversal
void DFS(vector<int> adj[], int V)
{
    std::vector<pair<int, int> > v;
    // Traversing for every vertex
    vector<bool> visited(V, false);
    for (int u = 1; u <= V; u++) {
        if (visited[u] == false) {
            int componentMax = u;
            int componentMin = u;
 
            DFSUtil(u, adj, visited,
                    componentMin, componentMax);
 
            // Storing maximum and minimum
            // values of each component
            v.push_back({ componentMin,
                          componentMax });
        }
    }
 
    bool check = isValid(v);
 
    if (check)
        cout << "Yes";
    else
        cout << "No";
 
    return;
}
 
// Driver code
int main()
{
    int N = 4, K = 3;
 
    vector<int> adj[N + 1];
 
    addEdge(adj, 1, 2);
    addEdge(adj, 2, 3);
    addEdge(adj, 3, 4);
 
    DFS(adj, N);
 
    return 0;
}

Java

// Java program of the 
// above approach
import java.util.*;
import java.lang.*;
 
class GFG{
     
static class pair
{
    int first, second;
    pair(int first, int second)
    {
        this.first = first;
        this.second = second;
    }
}
 
// Function to add edge into
// the graph
static void addEdge(ArrayList<ArrayList<Integer>> adj,
                    int u, int v)
{
    adj.get(u).add(v);
    adj.get(v).add(u);
}
 
static void DFSUtil(int u, 
                    ArrayList<ArrayList<Integer>> adj,
                    boolean[] visited,
                    int componentMin,
                    int componentMax)
{
    visited[u] = true;
 
    // Finding the maximum and
    // minimum values in each component
    componentMax = Math.max(componentMax, u);
    componentMin = Math.min(componentMin, u);
 
    for(int i = 0; i < adj.get(u).size(); i++)
        if (visited[adj.get(u).get(i)] == false)
            DFSUtil(adj.get(u).get(i), adj, visited,
                    componentMin, componentMax);
}
 
// Function for checking whether
// the given graph is valid or not
static boolean isValid(ArrayList<pair> v)
{
    int MAX = -1;
    boolean ans = false;
     
    // Checking for intersecting intervals
    for(pair i : v)
    {
         
        // If intersection is found
        if (i.first <= MAX)
        {
             
            // Graph is not valid
            ans = true;
        }
        MAX = Math.max(MAX, i.second);
    }
    return (ans == false ? true : false);
}
 
// Function for the DFS Traversal
static void DFS(ArrayList<ArrayList<Integer>> adj, 
                int V)
{
   ArrayList<pair> v = new ArrayList<>();
    
   // Traversing for every vertex
   boolean[] visited = new boolean[V + 1];
    
    for(int u = 1; u <= V; u++)
    {
        if (visited[u] == false) 
        {
            int componentMax = u;
            int componentMin = u;
 
            DFSUtil(u, adj, visited,
                    componentMin,
                    componentMax);
 
            // Storing maximum and minimum
            // values of each component
            v.add(new pair(componentMin,
                           componentMax));
        }
    }
 
    boolean check = isValid(v);
 
    if (check)
        System.out.println("Yes");
    else
        System.out.println("No");
 
    return;
}
 
// Driver code
public static void main (String[] args)
{
    int N = 4, K = 3;
     
    ArrayList<ArrayList<Integer>> adj = new ArrayList<>();
     
    for(int i = 0; i <= N + 1; i++)
        adj.add(new ArrayList<>());
     
    addEdge(adj, 1, 2);
    addEdge(adj, 2, 3);
    addEdge(adj, 3, 4);
     
    DFS(adj, N);
}
}
 
// This code is contributed by offbeat

Python3

# Python3 program of the
# above approach
 
# Function to add edge into
# the graph
def addEdge(adj, u, v):
 
    adj[u].append(v)
    adj[v].append(u)
    return adj
 
def DFSUtil(u, adj, visited, 
            componentMin, componentMax):
 
    visited[u] = True
 
    # Finding the maximum and
    # minimum values in each component
    componentMax = max(componentMax, u)
    componentMin = min(componentMin, u)
 
    for i in range(len(adj[u])):
        if (visited[adj[u][i]] == False):
            visited, componentMax, componentMin = DFSUtil(
                adj[u][i], adj, visited, componentMin, 
                componentMax)
             
    return visited, componentMax, componentMin
 
# Function for checking whether
# the given graph is valid or not
def isValid(v):
 
    MAX = -1
    ans = False
 
    # Checking for intersecting intervals
    for i in v:
        if len(i) != 2:
            continue
         
        # If intersection is found
        if (i[0] <= MAX):
 
            # Graph is not valid
            ans = True
 
        MAX = max(MAX, i[1])
 
    return (True if ans == False else False)
 
# Function for the DFS Traversal
def DFS(adj, V):
 
    v = [[]]
     
    # Traversing for every vertex
    visited = [False for i in range(V + 1)]
     
    for u in range(1, V + 1):
        if (visited[u] == False):
            componentMax = u
            componentMin = u
 
            visited, componentMax, componentMin = DFSUtil(
                u, adj, visited, componentMin,
                componentMax)
 
            # Storing maximum and minimum
            # values of each component
            v.append([componentMin, componentMax])
 
    check = isValid(v)
 
    if (check):
        print('Yes')
    else:
        print('No')
 
    return
 
# Driver code
if __name__=="__main__":
 
    N = 4
    K = 3
 
    adj = [[] for i in range(N + 1)]
 
    adj = addEdge(adj, 1, 2)
    adj = addEdge(adj, 2, 3)
    adj = addEdge(adj, 3, 4)
 
    DFS(adj, N)
 
# This code is contributed by rutvik_56

C#

// C# program of the 
// above approach
using System;
using System.Collections;
using System.Collections.Generic;
  
class GFG{
      
class pair
{
    public int first, second;
    public pair(int first, int second)
    {
        this.first = first;
        this.second = second;
    }
}
  
// Function to add edge into
// the graph
static void addEdge(ArrayList adj,
                    int u, int v)
{
    ((ArrayList)adj[u]).Add(v);
    ((ArrayList)adj[v]).Add(u);
}
  
static void DFSUtil(int u, ArrayList adj,
                    bool[] visited,
                    int componentMin,
                    int componentMax)
{
    visited[u] = true;
  
    // Finding the maximum and
    // minimum values in each component
    componentMax = Math.Max(componentMax, u);
    componentMin = Math.Min(componentMin, u);
  
    for(int i = 0; i < ((ArrayList)adj[u]).Count; i++)
        if (visited[(int)((ArrayList)adj[u])[i]] == false)
            DFSUtil((int)((ArrayList)adj[u])[i], adj, visited,
                    componentMin, componentMax);
}
  
// Function for checking whether
// the given graph is valid or not
static bool isValid(ArrayList v)
{
    int MAX = -1;
    bool ans = false;
      
    // Checking for intersecting intervals
    foreach(pair i in v)
    {
          
        // If intersection is found
        if (i.first <= MAX)
        {
              
            // Graph is not valid
            ans = true;
        }
        MAX = Math.Max(MAX, i.second);
    }
    return (ans == false ? true : false);
}
  
// Function for the DFS Traversal
static void DFS(ArrayList adj, 
                int V)
{
   ArrayList v = new ArrayList();
     
   // Traversing for every vertex
   bool[] visited = new bool[V + 1];
     
    for(int u = 1; u <= V; u++)
    {
        if (visited[u] == false) 
        {
            int componentMax = u;
            int componentMin = u;
  
            DFSUtil(u, adj, visited,
                    componentMin,
                    componentMax);
  
            // Storing maximum and minimum
            // values of each component
            v.Add(new pair(componentMin,
                           componentMax));
        }
    }
  
    bool check = isValid(v);
  
    if (check)
        Console.WriteLine("Yes");
    else
        Console.WriteLine("No");
  
    return;
}
  
// Driver code
public static void Main(string[] args)
{
    int N = 4;
      
    ArrayList adj = new ArrayList();
      
    for(int i = 0; i <= N + 1; i++)
        adj.Add(new ArrayList());
      
    addEdge(adj, 1, 2);
    addEdge(adj, 2, 3);
    addEdge(adj, 3, 4);
      
    DFS(adj, N);
}
}
 
// This code is contributed by pratham76

Javascript

<script>
 
// JavaScript program of the 
// above approach
      
class pair
{
    constructor(first, second)
    {
        this.first = first;
        this.second = second;
    }
}
  
// Function to add edge into
// the graph
function addEdge(adj,u, v)
{
    (adj[u]).push(v);
    (adj[v]).push(u);
}
  
function DFSUtil(u, adj, visited, componentMin, componentMax)
{
    visited[u] = true;
  
    // Finding the maximum and
    // minimum values in each component
    componentMax = Math.max(componentMax, u);
    componentMin = Math.min(componentMin, u);
  
    for(var i = 0; i < (adj[u]).length; i++)
        if (visited[(adj[u])[i]] == false)
            DFSUtil((adj[u])[i], adj, visited,
                    componentMin, componentMax);
}
  
// Function for checking whether
// the given graph is valid or not
function isValid(v)
{
    var MAX = -1;
    var ans = false;
      
    // Checking for intersecting intervals
    for(var i of v)
    {
          
        // If intersection is found
        if (i.first <= MAX)
        {
              
            // Graph is not valid
            ans = true;
        }
        MAX = Math.max(MAX, i.second);
    }
    return (ans == false ? true : false);
}
  
// Function for the DFS Traversal
function DFS(adj, V)
{
   var v = [];
     
   // Traversing for every vertex
   var visited = Array(V+1).fill(false);
     
    for(var u = 1; u <= V; u++)
    {
        if (visited[u] == false) 
        {
            var componentMax = u;
            var componentMin = u;
  
            DFSUtil(u, adj, visited,
                    componentMin,
                    componentMax);
  
            // Storing maximum and minimum
            // values of each component
            v.push(new pair(componentMin,
                           componentMax));
        }
    }
  
    var check = isValid(v);
  
    if (check)
        document.write("Yes");
    else
        document.write("No");
  
    return;
}
  
// Driver code
var N = 4;
  
var adj = [];
  
for(var i = 0; i <= N + 1; i++)
    adj.push(new Array());
  
addEdge(adj, 1, 2);
addEdge(adj, 2, 3);
addEdge(adj, 3, 4);
  
DFS(adj, N);
 
 
</script>

Output:

Yes

Time Complexity: O(N + E)
Auxiliary Space: O(N)

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