什么是歐拉路徑？

歐拉路徑是無向連通圖中的一條路徑留晚，該路徑經(jīng)過圖的每一條邊且僅經(jīng)過一次酵紫。如果路徑起點(diǎn)和終點(diǎn)相同，則稱“歐拉回路”错维。具有歐拉回路的圖稱“歐拉圖”奖地。

如何判斷圖中是否存在歐拉路徑？

由歐拉路徑的定義可知赋焕，若圖中存在歐拉路徑参歹，則該圖必是一個(gè)連通圖 (1)，其次隆判，圖中度數(shù)為奇數(shù)的點(diǎn)的個(gè)數(shù)必須為0或2 (2)犬庇，若度數(shù)為奇數(shù)的點(diǎn)的個(gè)數(shù)為0則是歐拉回路，若個(gè)數(shù)為2則是非歐拉回路的歐拉路徑在此題中稱為"Semi-Eulerian"侨嘀，其余情況均不是歐拉路徑臭挽。

原題

1126 Eulerian Path (25 分)

In graph theory, an Eulerian path is a path in a graph which visits every edge exactly once. Similarly, an Eulerian circuit is an Eulerian path which starts and ends on the same vertex. They were first discussed by Leonhard Euler while solving the famous Seven Bridges of Konigsberg problem in 1736. It has been proven that connected graphs with all vertices of even degree have an Eulerian circuit, and such graphs are called Eulerian. If there are exactly two vertices of odd degree, all Eulerian paths start at one of them and end at the other. A graph that has an Eulerian path but not an Eulerian circuit is called semi-Eulerian. (Cited from https://en.wikipedia.org/wiki/Eulerian_path)
Given an undirected graph, you are supposed to tell if it is Eulerian, semi-Eulerian, or non-Eulerian.

Input Specification:

Each input file contains one test case. Each case starts with a line containing 2 numbers N (≤ 500), and M, which are the total number of vertices, and the number of edges, respectively. Then M lines follow, each describes an edge by giving the two ends of the edge (the vertices are numbered from 1 to N).

Output Specification:

For each test case, first print in a line the degrees of the vertices in ascending order of their indices. Then in the next line print your conclusion about the graph -- either Eulerian, Semi-Eulerian, or Non-Eulerian. Note that all the numbers in the first line must be separated by exactly 1 space, and there must be no extra space at the beginning or the end of the line.

Sample Input 1:

7 12
5 7
1 2
1 3
2 3
2 4
3 4
5 2
7 6
6 3
4 5
6 4
5 6

Sample Output 1:

2 4 4 4 4 4 2
Eulerian

Sample Input 2:

6 10
1 2
1 3
2 3
2 4
3 4
5 2
6 3
4 5
6 4
5 6

Sample Output 2:

2 4 4 4 3 3
Semi-Eulerian

Sample Input 3:

5 8
1 2
2 5
5 4
4 1
1 3
3 2
3 4
5 3

Sample Output 3:

3 3 4 3 3
Non-Eulerian

AC代碼(C++)

#include <iostream>
#include <vector>
using namespace std;
int N, M, v1, v2, visited[505], degrees[505], judge = 1, cnt;
vector<int>graph[505];
void dfs(int d){
    visited[d] = 1;
    for(int i = 0; i < graph[d].size(); i++){
        if(!visited[graph[d][i]])dfs(graph[d][i]);
    }
}
int main(){
    scanf("%d%d", &N, &M);
    for(int i = 0; i < M; i++){
        scanf("%d%d", &v1, &v2);
        graph[v1].push_back(v2);
        graph[v2].push_back(v1);
        degrees[v1]++;
        degrees[v2]++;
    }
    dfs(1);
    for(int i = 1; i <= N; i++)
        if(!visited[i]){
            judge = 0;
            break;
        }
    for(int i = 1; i <= N; i++)
        if(i == 1)printf("%d", degrees[i]);
        else printf(" %d", degrees[i]);
    printf("\n");
    if(judge){
        for(int i = 1; i <= N; i++)
            if(degrees[i] % 2 == 1)cnt++;
        if(cnt == 0)printf("Eulerian\n");
        else if(cnt == 2) printf("Semi-Eulerian\n");
        else printf("Non-Eulerian\n");
    }else printf("Non-Eulerian\n");
    return 0;
}