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法学欧拉路径与欧拉回路
欧拉路径是一条路径，我们可以通过它精确访问每条边一次。我们可以多次使用相同的顶点。欧拉回路是一种特殊的欧拉路径。当欧拉路径的起始顶点也与该路径的结束顶点相连时，则称为欧拉回路。
为了检测路径和回路，我们必须遵循以下条件：
- 图必须是连通的。
- 当恰好有两个顶点具有奇数度时，它是一条欧拉路径。
- 现在，当无向图的任何顶点都没有奇数度时，它就是一个欧拉回路。
输入和输出
Input: Adjacency matrix of a graph. 0 1 1 1 0 1 0 1 0 0 1 1 0 0 0 1 0 0 0 1 0 0 0 1 0 Output: The graph has an Eulerian path.
算法
traverse(u, visited)
输入：起始节点 u 和已访问节点，用于标记哪个节点已被访问。
输出：遍历所有连通顶点。
Begin mark u as visited for all vertex v, if it is adjacent with u, do if v is not visited, then traverse(v, visited) done End
isConnected(graph)
输入：图。
输出：如果图是连通的，则返回 True。
Begin define visited array for all vertices u in the graph, do make all nodes unvisited traverse(u, visited) if any unvisited node is still remaining, then return false done return true End
isEulerian(Graph)
输入：给定的图。
输出：如果是非欧拉图则返回 0，如果具有欧拉路径则返回 1，如果找到欧拉回路则返回 2。
Begin if isConnected() is false, then return false define list of degree for each node oddDegree := 0 for all vertex i in the graph, do for all vertex j which are connected with i, do increase degree done if degree of vertex i is odd, then increase dooDegree done if oddDegree > 2, then return 0 if oddDegree = 0, then return 2 else return 1 End
示例
#include<iostream>
#include<vector>
#define NODE 5
using namespace std;
int graph[NODE][NODE] = {
{0, 1, 1, 1, 0},
{1, 0, 1, 0, 0},
{1, 1, 0, 0, 0},
{1, 0, 0, 0, 1},
{0, 0, 0, 1, 0}
};
/* int graph[NODE][NODE] = {
{0, 1, 1, 1, 1},
{1, 0, 1, 0, 0},
{1, 1, 0, 0, 0},
{1, 0, 0, 0, 1},
{1, 0, 0, 1, 0}
};
*/ //uncomment to check Euler Circuit
/* int graph[NODE][NODE] = {
{0, 1, 1, 1, 0},
{1, 0, 1, 1, 0},
{1, 1, 0, 0, 0},
{1, 1, 0, 0, 1},
{0, 0, 0, 1, 0}
};
*/ //Uncomment to check Non Eulerian Graph
void traverse(int u, bool visited[]) {
visited[u] = true; //mark v as visited
for(int v = 0; v<NODE; v++) {
if(graph[u][v]) {
if(!visited[v])
traverse(v, visited);
}
}
}
bool isConnected() {
bool *vis = new bool[NODE];
//for all vertex u as start point, check whether all nodes are visible or not
for(int u; u < NODE; u++) {
for(int i = 0; i<NODE; i++)
vis[i] = false; //initialize as no node is visited
traverse(u, vis);
for(int i = 0; i<NODE; i++) {
if(!vis[i]) //if there is a node, not visited by traversal, graph is not connected
return false;
}
}
return true;
}
int isEulerian() {
if(isConnected() == false) //when graph is not connected
return 0;
vector<int> degree(NODE, 0);
int oddDegree = 0;
for(int i = 0; i<NODE; i++) {
for(int j = 0; j<NODE; j++) {
if(graph[i][j])
degree[i]++; //increase degree, when connected edge found
}
if(degree[i] % 2 != 0) //when degree of vertices are odd
oddDegree++; //count odd degree vertices
}
if(oddDegree > 2) //when vertices with odd degree greater than 2
return 0;
return (oddDegree)?1:2; //when oddDegree is 0, it is Euler circuit, and when 2, it is Euler path
}
int main() {
int check;
check = isEulerian();
switch(check) {
case 0: cout << "The graph is not an Eulerian graph.";
break;
case 1: cout << "The graph has an Eulerian path.";
break;
case 2: cout << "The graph has a Eulerian circuit.";
break;
}
}输出
The graph has an Eulerian path.
更新于：2020年6月16日
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