Eulerian Path es un camino en el gráfico que visita cada borde exactamente una vez. El Circuito Euleriano es un Camino Euleriano que comienza y termina en el mismo vértice.
¿Cómo encontrar si un gráfico dado es Euleriano o no?
El problema es el mismo que la siguiente pregunta. “¿Es posible dibujar un gráfico dado sin levantar el lápiz del papel y sin trazar ninguno de los bordes más de una vez?”.
Un grafo se llama euleriano si tiene un ciclo euleriano y se llama semieuleriano si tiene un camino euleriano. El problema parece similar al camino hamiltoniano, que es un problema completo NP para un gráfico general. Afortunadamente, podemos encontrar si un gráfico dado tiene un Camino Euleriano o no en tiempo polinomial. De hecho, podemos encontrarlo en tiempo O(V+E).
A continuación se presentan algunas propiedades interesantes de los grafos no dirigidos con un camino y un ciclo eulerianos. Podemos usar estas propiedades para saber si un grafo es euleriano o no.
Ciclo euleriano: un gráfico no dirigido tiene un ciclo euleriano si las siguientes dos condiciones son verdaderas.
- Todos los vértices con grado distinto de cero están conectados. No nos importan los vértices con grado cero porque no pertenecen al ciclo o camino euleriano (solo consideramos todas las aristas).
- Todos los vértices tienen grado par.
Eulerian Path: un gráfico no dirigido tiene Eulerian Path si las dos condiciones siguientes son verdaderas.
- Igual que la condición (a) para el Ciclo Euleriano.
- Si cero o dos vértices tienen grado impar y todos los demás vértices tienen grado par. Tenga en cuenta que solo un vértice con grado impar no es posible en un gráfico no dirigido (la suma de todos los grados siempre es par en un gráfico no dirigido)
Tenga en cuenta que un gráfico sin bordes se considera euleriano porque no hay bordes para atravesar.
¿Como funciona esto?
En el camino euleriano, cada vez que visitamos un vértice v, caminamos a través de dos aristas no visitadas con un punto final como v. Por lo tanto, todos los vértices medios en el camino euleriano deben tener un grado par. Para el Ciclo Euleriano, cualquier vértice puede ser un vértice medio, por lo tanto, todos los vértices deben tener un grado par.
Implementación:
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 graph 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 vertices // 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 // Array of 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 graph 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 vertices // with zero degree Graph g5 = new Graph(3); g5.test(); } } // This code is contributed by Aakash Hasija
Python3
# 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 False return True '''The function returns one of the following values 0 --> If graph 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 vertices # 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 // Array of 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 graph 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 vertices // with zero degree Graph g5 = new Graph(3); g5.test(); } } // This code contributed by PrinciRaj1992
Javascript
<script> // A Javascript program to check if a given graph is Eulerian or not // This class represents an undirected graph using adjacency list // representation class Graph { // Constructor constructor(v) { this.V = v; this.adj = new Array(v); for (let i = 0; i < v; ++i) this.adj[i] = []; } // Function to add an edge into the graph addEdge(v,w) { this.adj[v].push(w);// Add w to v's list. this.adj[w].push(v); //The graph is undirected } // A function used by DFS DFSUtil(v,visited) { // Mark the current node as visited visited[v] = true; // Recur for all the vertices adjacent to this vertex for(let i of this.adj[v]) { let n = i; if (!visited[n]) this.DFSUtil(n, visited); } } // Method to check if all non-zero degree vertices are // connected. It mainly does DFS traversal starting from isConnected() { // Mark all the vertices as not visited let visited = new Array(this.V); let i; for (i = 0; i < this.V; i++) visited[i] = false; // Find a vertex with non-zero degree for (i = 0; i < this.V; i++) if (this.adj[i].length != 0) break; // If there are no edges in the graph, return true if (i == this.V) return true; // Start DFS traversal from a vertex with non-zero degree this.DFSUtil(i, visited); // Check if all non-zero degree vertices are visited for (i = 0; i < this.V; i++) if (visited[i] == false && this.adj[i].length > 0) return false; return true; } /* The function returns one of the following values 0 --> If graph is not Eulerian 1 --> If graph has an Euler path (Semi-Eulerian) 2 --> If graph has an Euler Circuit (Eulerian) */ isEulerian() { // Check if all non-zero degree vertices are connected if (this.isConnected() == false) return 0; // Count vertices with odd degree let odd = 0; for (let i = 0; i < this.V; i++) if (this.adj[i].length%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 test() { let res = this.isEulerian(); if (res == 0) document.write("graph is not Eulerian<br>"); else if (res == 1) document.write("graph has a Euler path<br>"); else document.write("graph has a Euler cycle<br>"); } } // Driver method // Let us create and test graphs shown in above figures let 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(); let 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(); let 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 let 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 vertices // with zero degree let g5 = new Graph(3); g5.test(); // This code is contributed by avanitrachhadiya2155 </script>
graph has a Euler path graph has a Euler cycle graph is not Eulerian graph has a Euler cycle graph has a Euler cycle
Complejidad temporal: O(V+E)
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Publicación traducida automáticamente
Artículo escrito por GeeksforGeeks-1 y traducido por Barcelona Geeks. The original can be accessed here. Licence: CCBY-SA