Problem:
Score : 800 points

Problem Statement

You are given a simple connected undirected graph consisting of N vertices and M edges.
The vertices are numbered 1 to N, and the edges are numbered 1 to M.

Edge i connects Vertex a_i and b_i bidirectionally.

Determine if three circuits (see Notes) can be formed using each of the edges exactly once.

Notes

A circuit is a cycle allowing repetitions of vertices but not edges.

Constraints

All values in input are integers.

1 \leq N,M \leq 10^{5}

1 \leq a_i, b_i \leq N

The given graph is simple and connected.

Input

Input is given from Standard Input in the following format:

N M
a_1 b_1
:
a_M b_M

Output

If three circuits can be formed using each of the edges exactly once, print Yes; if they cannot, print No.

Sample Input 1

7 9
1 2
1 3
2 3
1 4
1 5
4 5
1 6
1 7
6 7

Sample Output 1

Yes

Three circuits can be formed using each of the edges exactly once, as follows:

Sample Input 2

3 3
1 2
2 3
3 1

Sample Output 2

No

Three circuits are needed.

Sample Input 3

18 27
17 7
12 15
18 17
13 18
13 6
5 7
7 1
14 5
15 11
7 6
1 9
5 4
18 16
4 6
7 2
7 11
6 3
12 14
5 2
10 5
7 8
10 15
3 15
9 8
7 15
5 16
18 15

Sample Output 3

Yes

Solution in C++14 (GCC 5.4.1) (judge verdict: Wrong Answer)

#include<stdio.h>
#include<stdlib.h>
#include<string.h>
#include<math.h>
#include<algorithm>
#include<queue>
#include<set>
#include<map>
#include<iostream>
using namespace std;
#define re register
#define ll long long
inline int gi()
{
	int f=1,sum=0;char ch=getchar();
	while(ch>'9' || ch<'0'){if(ch=='-')f=-1;ch=getchar();}
	while(ch>='0' && ch<='9'){sum=(sum<<3)+(sum<<1)+ch-'0';ch=getchar();}
	return f*sum;
}
const int N=100010;
int d[N],n,m,vis[N];
set<int>se[N];
int main()
{
	n=gi();m=gi();
	for(int i=1;i<=m;i++)
	{
		int u=gi(),v=gi();
		d[u]++;d[v]++;
		se[u].insert(v);
		se[v].insert(u);
	}
	for(int i=1;i<=n;i++)if(d[i]&1)return puts("No"),0;
	int huan=0;
	for(int i=1;i<=n;i++)
	{
		if(se[i].size()!=2)continue;
		vis[i]=1;
		set<int>::iterator id;
		id=se[i].begin();
		int v1=*id;id++;int v2=*id;
		se[v1].erase(se[v1].find(i));
		se[v2].erase(se[v2].find(i));
		if(v1==v2)huan++;
		se[v1].insert(v2);
		se[v2].insert(v1);
	}
	int Del=0,Max=0;
	for(int i=1;i<=n;i++)
		if(!vis[i])Del++,Max=max(Max,(int)se[i].size());
	huan+=Max/2;
	if(huan>=3 || Del>=3)return puts("Yes"),0;
	puts("No");
	return 0;
}