检查给定的图是否表示星形拓扑

给定图形G ，任务是检查它是否表示星形拓扑。
下图显示了一种星形拓扑：

例子：

Input : Graph = 

Output : YES

Input : Graph =

Output : NO

如果满足以下三个条件，则V顶点的图表示星形拓扑：

1. 一个节点(也称为中央节点)的度数为V – 1。
2. 除中央节点外，所有节点的度数均为1。
3. 边数=顶点数– 1。

想法是遍历图形并检查其是否满足以上三个条件。如果是，则表示星形拓扑。
下面是上述方法的实现：

CPP

// CPP program to check if the given graph
// represents a Star Topology
#include 
 
using namespace std;
 
// A utility function to add an edge in an
// undirected graph.
void addEdge(vector adj[], int u, int v)
{
    adj[u].push_back(v);
    adj[v].push_back(u);
}
 
// A utility function to print the adjacency list
// representation of graph
void printGraph(vector adj[], int V)
{
    for (int v = 0; v < V; ++v) {
        cout << "\n Adjacency list of vertex "
             << v << "\n head ";
        for (auto x : adj[v])
            cout << "-> " << x;
        printf("\n");
    }
}
 
/* Function to return true if the graph represented
   by the adjacency list represents a Star topology
   else return false */
bool checkStarTopologyUtil(vector adj[], int V, int E)
{
 
    // Number of edges should be equal
    // to (Number of vertices - 1)
    if (E != (V - 1))
        return false;
 
    // a single node is termed as a star topology
    // having only a central node
    if (V == 1)
        return true;
 
    int* vertexDegree = new int[V + 1];
    memset(vertexDegree, 0, sizeof vertexDegree);
 
    // calculate the degree of each vertex
    for (int i = 1; i <= V; i++) {
        for (auto v : adj[i]) {
            vertexDegree[v]++;
        }
    }
 
    // countCentralNodes strores the count of nodes
    // with degree V - 1, which should be equal to 1
    // in case of star topology
    int countCentralNodes = 0, centralNode = 0;
 
    for (int i = 1; i <= V; i++) {
        if (vertexDegree[i] == (V - 1)) {
            countCentralNodes++;
            // Store the index of the central node
            centralNode = i;
        }
    }
 
    // there should be only one central node
    // in the star topology
    if (countCentralNodes != 1)
        return false;
 
    for (int i = 1; i <= V; i++) {
        // except for the central node
        // check if all other nodes have
        // degree 1, if not return false
        if (i == centralNode)
            continue;
        if (vertexDegree[i] != 1) {
            return false;
        }
    }
 
    // if all three necessary
    // conditions as discussed,
    // satisfy return true
    return true;
}
 
// Function to check if the graph
// represents a Star topology
void checkStarTopology(vector adj[], int V, int E)
{
    bool isStar = checkStarTopologyUtil(adj, V, E);
    if (isStar) {
        cout << "YES" << endl;
    }
    else {
        cout << "NO" << endl;
    }
}
 
// Driver code
int main()
{
    // Graph 1
    int V = 5, E = 4;
    vector adj1[V + 1];
    addEdge(adj1, 1, 2);
    addEdge(adj1, 1, 3);
    addEdge(adj1, 1, 4);
    addEdge(adj1, 1, 5);
    checkStarTopology(adj1, V, E);
 
    // Graph 2
    V = 5, E = 4;
    vector adj2[V + 1];
    addEdge(adj2, 1, 2);
    addEdge(adj2, 1, 3);
    addEdge(adj2, 3, 4);
    addEdge(adj2, 4, 5);
    checkStarTopology(adj2, V, E);
 
    return 0;
}

Java

// Java program to check if the given graph
// represents a star topology
import java.io.*;
import java.util.*;
 
class GFG
{
 
  // A utility function to add an edge in an
  // undirected graph.
  static void addEdge(ArrayList> adj, int u, int v)
  {
    adj.get(u).add(v);
    adj.get(v).add(u);
  }
 
  // A utility function to print the adjacency list
  // representation of graph
  static void printGraph(ArrayList> adj, int V)
  {
    for (int v = 0; v < V; ++v)
    {
      System.out.print("\n Adjacency list of vertex " +
                       v + "\n head ");
      for (int x : adj.get(v))
      {
        System.out.print( "-> " + x);
      }
      System.out.println();
    }
  }
 
  /* Function to return true if the graph represented
    by the adjacency list represents a Star topology
    else return false */
  static boolean checkStarTopologyUtil(ArrayList> adj, int V, int E)
  {
    // Number of edges should be equal
    // to (Number of vertices - 1)
    if (E != (V - 1))
    {
      return false;
    }
 
    // a single node is termed as a star topology
    // having only a central node
    if (V == 1)
    {
      return true;
    }
    int[] vertexDegree = new int[V + 1];
 
    // calculate the degree of each vertex
    for (int i = 1; i <= V; i++)
    {
      for (int v : adj.get(i))
      {
        vertexDegree[v]++;
      }
    }
 
    // countCentralNodes strores the count of nodes 
    // with degree V - 1, which should be equal to 1
    // in case of star topology
    int countCentralNodes = 0, centralNode = 0;    
    for (int i = 1; i <= V; i++)
    {
      if (vertexDegree[i] == (V - 1))
      {
        countCentralNodes++;
 
        // Store the index of the central node
        centralNode = i;
      }
    }
 
    // there should be only one central node
    // in the star topology
    if (countCentralNodes != 1)
      return false;  
    for (int i = 1; i <= V; i++)
    {
 
      // except for the central node 
      // check if all other nodes have
      // degree 1, if not return false
      if (i == centralNode)
        continue;
      if (vertexDegree[i] != 1)
      {
        return false;
      }
    }
 
    // if all three necessary
    // conditions as discussed,
    // satisfy return true
    return true;
  }
 
  // Function to check if the graph 
  // represents a Star topology
  static void checkStarTopology(ArrayList> adj, int V, int E)
  {
    boolean isStar = checkStarTopologyUtil(adj, V, E);
    if (isStar)
    {
      System.out.println("YES");
    }
    else
    {
      System.out.println("NO");
    }
  }
 
  // Driver code
  public static void main (String[] args)
  {
 
    // Graph 1
    int V = 5, E = 4;
    ArrayList> adj1 =
      new ArrayList>();
    for(int i = 0; i < V + 1; i++)
    {
      adj1.add(new ArrayList());
    }
    addEdge(adj1, 1, 2);
    addEdge(adj1, 1, 3);
    addEdge(adj1, 1, 4);
    addEdge(adj1, 1, 5);
    checkStarTopology(adj1, V, E);
 
    // Graph 2
    V = 5;
    E = 4;
    ArrayList> adj2 =
      new ArrayList>();
    for(int i = 0; i < (V + 1); i++)
    {
      adj2.add(new ArrayList());
    }
    addEdge(adj2, 1, 2);
    addEdge(adj2, 1, 3);
    addEdge(adj2, 3, 4);
    addEdge(adj2, 4, 5);
    checkStarTopology(adj2, V, E);
  }
}
 
// This code is contributed by rag2127

Python3

# Python3 program to check if the given graph
# represents a star topology
 
# A utility function to add an edge in an
# undirected graph.
def addEdge(adj, u, v):
    adj[u].append(v)
    adj[v].append(u)
 
# A utility function to print the adjacency list
# representation of graph
def printGraph(adj, V):
 
    for v in range(V):
        print("Adjacency list of vertex ",v,"\n head ")
        for x in adj[v]:
            print("-> ",x,end=" ")
        printf()
 
# /* Function to return true if the graph represented
#    by the adjacency list represents a star topology
#    else return false */
def checkStarTopologyUtil(adj, V, E):
 
    # Number of edges should be equal
    # to (Number of vertices - 1)
    if (E != (V - 1)):
        return False
 
    # a single node is termed as a bus topology
    if (V == 1):
        return True
 
    vertexDegree = [0]*(V + 1)
 
    # calculate the degree of each vertex
    for i in range(V+1):
        for v in adj[i]:
            vertexDegree[v] += 1
 
    # countCentralNodes strores the count of nodes
    # with degree V - 1, which should be equal to 1
    # in case of star topology
    countCentralNodes = 0
    centralNode = 0
 
    for i in range(1, V + 1):
        if (vertexDegree[i] == (V - 1)):
            countCentralNodes += 1
            # Store the index of the central node
            centralNode = i
 
    # there should be only one central node
    # in the star topology
    if (countCentralNodes != 1):
        return False
 
    for i in range(1, V + 1):
        # except for the central node
        # check if all other nodes have
        # degree 1, if not return false
        if (i == centralNode):
            continue
        if (vertexDegree[i] != 1):
            return False
 
    # if all three necessary
    # conditions as discussed,
    # satisfy return true
    return True
 
# Function to check if the graph represents a bus topology
def checkStarTopology(adj, V, E):
 
    isStar = checkStarTopologyUtil(adj, V, E)
    if (isStar):
        print("YES")
 
    else:
        print("NO" )
 
# Driver code
 
# Graph 1
V, E = 5, 4
adj1=[[] for i in range(V + 1)]
addEdge(adj1, 1, 2)
addEdge(adj1, 1, 3)
addEdge(adj1, 1, 4)
addEdge(adj1, 1, 5)
checkStarTopology(adj1, V, E)
 
# Graph 2
V, E = 4, 4
adj2=[[] for i in range(V + 1)]
addEdge(adj2, 1, 2)
addEdge(adj2, 1, 3)
addEdge(adj2, 3, 4)
addEdge(adj2, 4, 2)
checkStarTopology(adj2, V, E)
 
# This code is contribued by mohit kumar 29

C#

// C# program to check if the given graph
// represents a star topology
using System;
using System.Collections.Generic;
class GFG
{
 
  // A utility function to add an edge in an
  // undirected graph.
  static void addEdge(List> adj, int u, int v)
  {
    adj[u].Add(v);
    adj[v].Add(u);
  }
 
  // A utility function to print the adjacency list
  // representation of graph
  static void printGraph(List> adj, int V)
  {
    for (int v = 0; v < V; ++v)
    {
      Console.WriteLine("\n Adjacency list of vertex " + v + "\n head ");
      foreach (int x in adj[v])
      {
        Console.Write( "-> " + x);
      }
      Console.WriteLine();
    }
  }
 
  /* Function to return true if the graph represented
    by the adjacency list represents a Star topology
    else return false */
  static bool checkStarTopologyUtil(List> adj, int V, int E)
  {
    // Number of edges should be equal
    // to (Number of vertices - 1)
    if (E != (V - 1))
    {
      return false;
    }
 
    // a single node is termed as a bus topology
    if (V == 1)
    {
      return true;
    }
    int[] vertexDegree = new int[V + 1];
 
    // calculate the degree of each vertex
    for (int i = 1; i <= V; i++)
    {
      foreach (int v in adj[i])
      {
        vertexDegree[v]++;
      }
    }
 
    // countCentralNodes strores the count of nodes 
    // with degree V - 1, which should be equal to 1
    // in case of star topology
    int countCentralNodes = 0, centralNode = 0;    
    for (int i = 1; i <= V; i++)
    {
      if (vertexDegree[i] == (V - 1))
      {
        countCentralNodes++;
 
        // Store the index of the central node
        centralNode = i;
      }
    }
 
    // there should be only one central node
    // in the star topology
    if (countCentralNodes != 1)
      return false;  
    for (int i = 1; i <= V; i++)
    {
 
      // except for the central node 
      // check if all other nodes have
      // degree 1, if not return false
      if (i == centralNode)
        continue;
      if (vertexDegree[i] != 1)
      {
        return false;
      }
    }
 
    // if all three necessary
    // conditions as discussed,
    // satisfy return true
    return true;
  }
 
  // Function to check if the graph 
  // represents a Star topology
  static void checkStarTopology(List> adj, int V, int E)
  {
    bool isStar = checkStarTopologyUtil(adj, V, E);
    if (isStar)
    {
      Console.WriteLine("YES");
    }
    else
    {
      Console.WriteLine("NO");
    }
  }
 
  // Driver code
  static public void Main ()
  {
 
    // Graph 1
    int V = 5, E = 4;
    List> adj1 = new List>();
    for(int i = 0; i < V + 1; i++)
    {
      adj1.Add(new List());
    }
    addEdge(adj1, 1, 2);
    addEdge(adj1, 1, 3);
    addEdge(adj1, 1, 4);
    addEdge(adj1, 1, 5);
    checkStarTopology(adj1, V, E);
 
    // Graph 2
    V = 5;
    E = 4;
    List> adj2 = new List>();
    for(int i = 0; i < V + 1; i++)
    {
      adj2.Add(new List());
    }
    addEdge(adj2, 1, 2);
    addEdge(adj2, 1, 3);
    addEdge(adj2, 3, 4);
    addEdge(adj2, 4, 4);
    checkStarTopology(adj2, V, E);
  }
}
 
// This code is contributed by avanitrachhadiya2155

输出：

YES
NO

时间复杂度： O(V + E)，其中V和E分别是图形中的顶点和边数。

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