1126 Eulerian Path (25 分) 欧拉回路判断

素颜马尾好姑娘i 2024-04-18 10:42 39阅读 0赞

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][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

乍一眼很简单，提交只有20分，嘤嘤嘤

原来是没有判断是否联通，大意了喵

#include<stdio.h>
    #include<string.h>
    #include<math.h>
    #include<iostream>
    #include<string>
    #include<sstream>
    #include<algorithm>
    #include<map>
    #include<set>
    #include<queue>
    #include<stack>
    #include<vector>
    using namespace std;
    #define inf 0x3f3f3f3f
    #define LL long long
    vector<int> v[505];
    bool vis[505];
    int sum;
    void dfs(int root)
    {
    	vis[root]=true;
    	sum++;
    	for(int i=0;i<v[root].size();i++)
    	{
    		if(vis[v[root][i]]==false)
    		dfs(v[root][i]);
    	}	
    }
    int main()
    {
    	int n,m,i,j,x,y,f=0,a[505]={0};
    	scanf("%d%d",&n,&m);
    	for(i=0;i<m;i++)
    	{
    		scanf("%d%d",&x,&y);
    		a[x]++,a[y]++;
    		v[x].push_back(y);
    		v[y].push_back(x);
    	}
    	dfs(1);
    	for(i=1;i<=n;i++)
    	{
    		if(a[i]%2)f++;
    		if(i!=1)printf(" ");
    		printf("%d",a[i]);
    	}
    	if(sum!=n)printf("\nNon-Eulerian\n");
    	else if(f==0)printf("\nEulerian\n");
    	else if(f==2)printf("\nSemi-Eulerian\n");
    	else printf("\nNon-Eulerian\n");
    	
    }

[https_en.wikipedia.org_wiki_Eulerian_path]: https://en.wikipedia.org/wiki/Eulerian_path