本文概述 欧拉路径是图形中的一条路径, 该路径恰好一次访问每个边。欧拉回路是一条始于和终止于同一顶点的欧拉路径。

文章图片

文章图片

文章图片
如何查找给定图是否为欧拉图？
问题与以下问题相同。 "有可能画出给定的图形而无需从纸上抬起铅笔, 也不必多次描画任何边缘。"
如果图形具有欧拉循环, 则称为欧拉曲线；如果图形具有欧拉路径, 则称为半欧拉曲线。问题似乎类似于哈密??顿路径这是一般图的NP完全问题。幸运的是, 我们可以发现给定图在多项式时间内是否具有欧拉路径。实际上, 我们可以在O(V + E)时间找到它。
以下是具有欧拉路径和循环的无向图的一些有趣特性。我们可以使用这些属性来查找图是否为欧拉图。
欧拉循环
如果满足以下两个条件, 则无向图具有欧拉循环。
….a)所有非零度的顶点都已连接。我们不在乎零度的顶点, 因为它们不属于欧拉循环或路径(我们仅考虑所有边)。
….b)所有顶点都具有偶数度。
欧拉路径
如果满足以下两个条件, 则无向图具有欧拉路径。
….a)与欧拉循环的条件(a)相同
….b)如果零个或两个顶点具有奇数度, 而所有其他顶点具有偶数度。请注意, 在无向图中, 只有一个具有奇数度的顶点是不可能的(在无向图中, 所有度的总和始终是偶数)
请注意, 没有边的图被视为欧拉图, 因为没有要遍历的边。
这是如何运作的？
【算法设计（计算无向图的欧拉路径和回路（））】在欧拉路径中, 每次访问顶点v时, 我们都会走过两个未访问的边缘, 且端点的端点为v。因此, 欧拉路径中的所有中间顶点都必须具有均匀度。对于欧拉循环, 任何顶点都可以是中间顶点, 因此所有顶点都必须具有偶数度。
C ++

// A C++ program to check if a given graph is Eulerian or not #include< iostream> #include < list> using namespace std; // A class that represents an undirected graph class Graph { int V; // No. of vertices list< int > *adj; // A dynamic array of adjacency lists public : // Constructor and destructor Graph( int V){ this -> V = V; adj = new list< int > [V]; } ~Graph() { delete [] adj; } // To avoid memory leak// function to add an edge to graph void addEdge( int v, int w); // Method to check if this graph is Eulerian or not int isEulerian(); // Method to check if all non-zero degree vertices are connected bool isConnected(); // Function to do DFS starting from v. Used in isConnected(); void DFSUtil( int v, bool visited[]); }; void Graph::addEdge( int v, int w) { adj[v].push_back(w); adj[w].push_back(v); // Note: the graph is undirected }void Graph::DFSUtil( 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]) DFSUtil(*i, visited); }// Method to check if all non-zero degree vertices are connected. // It mainly does DFS traversal starting from bool Graph::isConnected() { // Mark all the vertices as not visited bool visited[V]; int i; for (i = 0; i < V; i++) visited[i] = false ; // Find a vertex with non-zero degree for (i = 0; i < V; i++) if (adj[i].size() != 0) break ; // If there are no edges in the graph, return true if (i == V) return true ; // Start DFS traversal from a vertex with non-zero degree DFSUtil(i, visited); // Check if all non-zero degree vertices are visited for (i = 0; i < V; i++) if (visited[i] == false & & adj[i].size() > 0) return false ; return true ; }/* The function returns one of the following values 0 --> If grpah is not Eulerian 1 --> If graph has an Euler path (Semi-Eulerian) 2 --> If graph has an Euler Circuit (Eulerian)*/ int Graph::isEulerian() { // Check if all non-zero degree vertices are connected if (isConnected() == false ) return 0; // Count vertices with odd degree int odd = 0; for ( int i = 0; i < V; i++) if (adj[i].size() & 1) odd++; // If count is more than 2, then graph is not Eulerian if (odd > 2) return 0; // If odd count is 2, then semi-eulerian. // If odd count is 0, then eulerian // Note that odd count can never be 1 for undirected graph return (odd)? 1 : 2; }// Function to run test cases void test(Graph & g) { int res = g.isEulerian(); if (res == 0) cout < < "graph is not Eulerian\n" ; else if (res == 1) cout < < "graph has a Euler path\n" ; else cout < < "graph has a Euler cycle\n" ; }// Driver program to test above function int main() { // Let us create and test graphs shown in above figures Graph g1(5); g1.addEdge(1, 0); g1.addEdge(0, 2); g1.addEdge(2, 1); g1.addEdge(0, 3); g1.addEdge(3, 4); test(g1); Graph g2(5); g2.addEdge(1, 0); g2.addEdge(0, 2); g2.addEdge(2, 1); g2.addEdge(0, 3); g2.addEdge(3, 4); g2.addEdge(4, 0); test(g2); Graph g3(5); g3.addEdge(1, 0); g3.addEdge(0, 2); g3.addEdge(2, 1); g3.addEdge(0, 3); g3.addEdge(3, 4); g3.addEdge(1, 3); test(g3); // Let us create a graph with 3 vertices // connected in the form of cycle Graph g4(3); g4.addEdge(0, 1); g4.addEdge(1, 2); g4.addEdge(2, 0); test(g4); // Let us create a graph with all veritces // with zero degree Graph g5(3); test(g5); return 0; }

Java

// A Java program to check if a given graph is Eulerian or not import java.io.*; import java.util.*; import java.util.LinkedList; // This class represents an undirected graph using adjacency list // representation class Graph { private int V; // No. of vertices// Arrayof lists for Adjacency List Representation private LinkedList< Integer> adj[]; // Constructor Graph( int v) { V = v; adj = new LinkedList[v]; for ( int i= 0 ; i< v; ++i) adj[i] = new LinkedList(); }//Function to add an edge into the graph void addEdge( int v, int w) { adj[v].add(w); // Add w to v's list. adj[w].add(v); //The graph is undirected }// A function used by DFS void DFSUtil( int v, boolean visited[]) { // Mark the current node as visited visited[v] = true ; // Recur for all the vertices adjacent to this vertex Iterator< Integer> i = adj[v].listIterator(); while (i.hasNext()) { int n = i.next(); if (!visited[n]) DFSUtil(n, visited); } }// Method to check if all non-zero degree vertices are // connected. It mainly does DFS traversal starting from boolean isConnected() { // Mark all the vertices as not visited boolean visited[] = new boolean [V]; int i; for (i = 0 ; i < V; i++) visited[i] = false ; // Find a vertex with non-zero degree for (i = 0 ; i < V; i++) if (adj[i].size() != 0 ) break ; // If there are no edges in the graph, return true if (i == V) return true ; // Start DFS traversal from a vertex with non-zero degree DFSUtil(i, visited); // Check if all non-zero degree vertices are visited for (i = 0 ; i < V; i++) if (visited[i] == false & & adj[i].size() > 0 ) return false ; return true ; }/* The function returns one of the following values 0 --> If grpah is not Eulerian 1 --> If graph has an Euler path (Semi-Eulerian) 2 --> If graph has an Euler Circuit (Eulerian)*/ int isEulerian() { // Check if all non-zero degree vertices are connected if (isConnected() == false ) return 0 ; // Count vertices with odd degree int odd = 0 ; for ( int i = 0 ; i < V; i++) if (adj[i].size()% 2 != 0 ) odd++; // If count is more than 2, then graph is not Eulerian if (odd > 2 ) return 0 ; // If odd count is 2, then semi-eulerian. // If odd count is 0, then eulerian // Note that odd count can never be 1 for undirected graph return (odd== 2 )? 1 : 2 ; }// Function to run test cases void test() { int res = isEulerian(); if (res == 0 ) System.out.println( "graph is not Eulerian" ); else if (res == 1 ) System.out.println( "graph has a Euler path" ); else System.out.println( "graph has a Euler cycle" ); }// Driver method public static void main(String args[]) { // Let us create and test graphs shown in above figures Graph g1 = new Graph( 5 ); g1.addEdge( 1 , 0 ); g1.addEdge( 0 , 2 ); g1.addEdge( 2 , 1 ); g1.addEdge( 0 , 3 ); g1.addEdge( 3 , 4 ); g1.test(); 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( 4 , 0 ); g2.test(); Graph g3 = new Graph( 5 ); g3.addEdge( 1 , 0 ); g3.addEdge( 0 , 2 ); g3.addEdge( 2 , 1 ); g3.addEdge( 0 , 3 ); g3.addEdge( 3 , 4 ); g3.addEdge( 1 , 3 ); g3.test(); // Let us create a graph with 3 vertices // connected in the form of cycle Graph g4 = new Graph( 3 ); g4.addEdge( 0 , 1 ); g4.addEdge( 1 , 2 ); g4.addEdge( 2 , 0 ); g4.test(); // Let us create a graph with all veritces // with zero degree Graph g5 = new Graph( 3 ); g5.test(); } } // This code is contributed by Aakash Hasija

python

# Python program to check if a given graph is Eulerian or not #Complexity : O(V+E)from collections import defaultdict#This class represents a undirected graph using adjacency list representation class Graph:def __init__( self , vertices): self .V = vertices #No. of vertices self .graph = defaultdict( list ) # default dictionary to store graph# function to add an edge to graph def addEdge( self , u, v): self .graph[u].append(v) self .graph[v].append(u)#A function used by isConnected def DFSUtil( self , v, visited): # Mark the current node as visited visited[v] = True#Recur for all the vertices adjacent to this vertex for i in self .graph[v]: if visited[i] = = False : self .DFSUtil(i, visited)'''Method to check if all non-zero degree vertices are connected. It mainly does DFS traversal starting from node with non-zero degree''' def isConnected( self ):# Mark all the vertices as not visited visited = [ False ] * ( self .V)#Find a vertex with non-zero degree for i in range ( self .V): if len ( self .graph[i]) > 1 : break# If there are no edges in the graph, return true if i = = self .V - 1 : return True# Start DFS traversal from a vertex with non-zero degree self .DFSUtil(i, visited)# Check if all non-zero degree vertices are visited for i in range ( self .V): if visited[i] = = False and len ( self .graph[i]) > 0 : return Falsereturn True'''The function returns one of the following values 0 --> If grpah is not Eulerian 1 --> If graph has an Euler path (Semi-Eulerian) 2 --> If graph has an Euler Circuit (Eulerian)''' def isEulerian( self ): # Check if all non-zero degree vertices are connected if self .isConnected() = = False : return 0 else : #Count vertices with odd degree odd = 0 for i in range ( self .V): if len ( self .graph[i]) % 2 ! = 0 : odd + = 1'''If odd count is 2, then semi-eulerian. If odd count is 0, then eulerian If count is more than 2, then graph is not Eulerian Note that odd count can never be 1 for undirected graph''' if odd = = 0 : return 2 elif odd = = 2 : return 1 elif odd > 2 : return 0# Function to run test cases def test( self ): res = self .isEulerian() if res = = 0 : print "graph is not Eulerian" elif res = = 1 : print "graph has a Euler path" else : print "graph has a Euler cycle"#Let us create and test graphs shown in above figures g1 = Graph( 5 ); g1.addEdge( 1 , 0 ) g1.addEdge( 0 , 2 ) g1.addEdge( 2 , 1 ) g1.addEdge( 0 , 3 ) g1.addEdge( 3 , 4 ) g1.test()g2 = Graph( 5 ) g2.addEdge( 1 , 0 ) g2.addEdge( 0 , 2 ) g2.addEdge( 2 , 1 ) g2.addEdge( 0 , 3 ) g2.addEdge( 3 , 4 ) g2.addEdge( 4 , 0 ) g2.test(); g3 = Graph( 5 ) g3.addEdge( 1 , 0 ) g3.addEdge( 0 , 2 ) g3.addEdge( 2 , 1 ) g3.addEdge( 0 , 3 ) g3.addEdge( 3 , 4 ) g3.addEdge( 1 , 3 ) g3.test()#Let us create a graph with 3 vertices # connected in the form of cycle g4 = Graph( 3 ) g4.addEdge( 0 , 1 ) g4.addEdge( 1 , 2 ) g4.addEdge( 2 , 0 ) g4.test()# Let us create a graph with all veritces # with zero degree g5 = Graph( 3 ) g5.test()#This code is contributed by Neelam Yadav

C#

// A C# program to check if a given graph is Eulerian or not using System; using System.Collections.Generic; // This class represents an undirected graph using adjacency list // representation public class Graph { private int V; // No. of vertices// Arrayof lists for Adjacency List Representation private List< int > []adj; // Constructor Graph( int v) { V = v; adj = new List< int > [v]; for ( int i=0; i< v; ++i) adj[i] = new List< int > (); }//Function to add an edge into the graph void addEdge( int v, int w) { adj[v].Add(w); // Add w to v's list. adj[w].Add(v); //The graph is undirected }// A function used by DFS void DFSUtil( int v, bool []visited) { // Mark the current node as visited visited[v] = true ; // Recur for all the vertices adjacent to this vertex foreach ( int i in adj[v]){ int n = i; if (!visited[n]) DFSUtil(n, visited); } }// Method to check if all non-zero degree vertices are // connected. It mainly does DFS traversal starting from bool isConnected() { // Mark all the vertices as not visited bool []visited = new bool [V]; int i; for (i = 0; i < V; i++) visited[i] = false ; // Find a vertex with non-zero degree for (i = 0; i < V; i++) if (adj[i].Count != 0) break ; // If there are no edges in the graph, return true if (i == V) return true ; // Start DFS traversal from a vertex with non-zero degree DFSUtil(i, visited); // Check if all non-zero degree vertices are visited for (i = 0; i < V; i++) if (visited[i] == false & & adj[i].Count > 0) return false ; return true ; }/* The function returns one of the following values 0 --> If grpah is not Eulerian 1 --> If graph has an Euler path (Semi-Eulerian) 2 --> If graph has an Euler Circuit (Eulerian)*/ int isEulerian() { // Check if all non-zero degree vertices are connected if (isConnected() == false ) return 0; // Count vertices with odd degree int odd = 0; for ( int i = 0; i < V; i++) if (adj[i].Count%2!=0) odd++; // If count is more than 2, then graph is not Eulerian if (odd > 2) return 0; // If odd count is 2, then semi-eulerian. // If odd count is 0, then eulerian // Note that odd count can never be 1 for undirected graph return (odd==2)? 1 : 2; }// Function to run test cases void test() { int res = isEulerian(); if (res == 0) Console.WriteLine( "graph is not Eulerian" ); else if (res == 1) Console.WriteLine( "graph has a Euler path" ); else Console.WriteLine( "graph has a Euler cycle" ); }// Driver method public static void Main(String []args) { // Let us create and test graphs shown in above figures Graph g1 = new Graph(5); g1.addEdge(1, 0); g1.addEdge(0, 2); g1.addEdge(2, 1); g1.addEdge(0, 3); g1.addEdge(3, 4); g1.test(); 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(4, 0); g2.test(); Graph g3 = new Graph(5); g3.addEdge(1, 0); g3.addEdge(0, 2); g3.addEdge(2, 1); g3.addEdge(0, 3); g3.addEdge(3, 4); g3.addEdge(1, 3); g3.test(); // Let us create a graph with 3 vertices // connected in the form of cycle Graph g4 = new Graph(3); g4.addEdge(0, 1); g4.addEdge(1, 2); g4.addEdge(2, 0); g4.test(); // Let us create a graph with all veritces // with zero degree Graph g5 = new Graph(3); g5.test(); } }// This code contributed by PrinciRaj1992

输出如下：

graph has a Euler path graph has a Euler cycle graph is not Eulerian graph has a Euler cycle graph has a Euler cycle

时间复杂度：O(V + E)
下一篇文章：
有向图的欧拉路径和电路。
Fleury的算法可以打印欧拉路径或电路？
参考文献：
http://en.wikipedia.org/wiki/Eulerian_path
如果发现任何不正确的地方, 或者你想分享有关上述主题的更多信息, 请发表评论

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