PAT 1126 Eulerian Path

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

 思路：就是找出每個節點的度為偶數的節點個數；如果度全為偶數，就是歐拉圖， 如果只有兩個點的度不是偶數就是半歐拉圖， 否則不是歐拉圖；

注意點：歐拉圖或者半歐拉圖的前提是圖是連通的

#include<iostream>
#include<vector>
using namespace std;
vector<int> vis(505, false);
vector<int> v[505];
void dfs(int node){
  vis[node] = true;
  for(int i=0; i<v[node].size(); i++)
   if(!vis[v[node][i]]) dfs(v[node][i]);
}
int main(){
  long n, k, i;
  cin>>n>>k;
  
  for(i=0; i<k; i++){
    int a, b;
    cin>>a>>b;
    v[a].push_back(b);
    v[b].push_back(a);
  }
  vector<int> num(505,0);
  int odd=0, even=0;
  for(i=1; i<=n; i++){
    num[i] = v[i].size();
    if(num[i]%2==0) even++;
    else odd++;
  }
  for(i=1; i<=n; i++){
    if(i==1) cout<<num[i];
    else cout<<" "<<num[i];
  }
  cout<<endl;
  int cnt=0;
  for(i=1; i<=n; i++){
    if(!vis[i]){
      dfs(i);
      cnt++;
    }
  }
  if(cnt!=1) cout<<"Non-Eulerian"<<endl;
  else{
    if(even==n) cout<<"Eulerian"<<endl;
    else if(odd==2) cout<<"Semi-Eulerian"<<endl;
    else cout<<"Non-Eulerian"<<endl;
  }
  
  return 0;
}