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: Accepted)

#include<bits/stdc++.h>
namespace my_std{
	using namespace std;
	#define pii pair<int,int>
	#define fir first
	#define sec second
	#define MP make_pair
	#define rep(i,x,y) for (int i=(x);i<=(y);i++)
	#define drep(i,x,y) for (int i=(x);i>=(y);i--)
	#define go(x) for (int i=head[x];i;i=edge[i].nxt)
	#define templ template<typename T>
	#define sz 101010
	typedef long long ll;
	typedef double db;
	templ inline bool chkmax(T &x,T y){return x<y?x=y,1:0;}
	templ inline bool chkmin(T &x,T y){return x>y?x=y,1:0;}
	templ inline void read(T& t)
	{
		t=0;char f=0,ch=getchar();double d=0.1;
		while(ch>'9'||ch<'0') f|=(ch=='-'),ch=getchar();
		while(ch<='9'&&ch>='0') t=t*10+ch-48,ch=getchar();
		if(ch=='.'){ch=getchar();while(ch<='9'&&ch>='0') t+=d*(ch^48),d*=0.1,ch=getchar();}
		t=(f?-t:t);
	}
	template<typename T,typename... Args>inline void read(T& t,Args&... args){read(t); read(args...);}
}
using namespace my_std;

int n,m;
struct hh{int t,nxt;}edge[sz<<1];
bool vis[sz];
int head[sz],ecnt;
int deg[sz];
void make_edge(int f,int t)
{
	edge[++ecnt]=(hh){t,head[f]};
	head[f]=ecnt;
	edge[++ecnt]=(hh){f,head[t]};
	head[t]=ecnt;
	++deg[f],++deg[t];
}

void out(string s){cout<<s;exit(0);}

int p1,p2;
int e;
int dfs(int x,bool t)
{
	if (x==p1&&!t) return 1;
	if (x==p2) return ++e,2; 
	#define v edge[i].t
	go(x) if (!vis[(i+1)>>1])
	{
		vis[(i+1)>>1]=1;
		int r=dfs(v,0);
		if (r==1&&x!=p1) return 1;
		if (r==2&&x!=p1) return 2;
	}
	#undef v
	return 0;
}

int main()
{
	read(n,m);
	int x,y;
	rep(i,1,m) read(x,y),make_edge(x,y);
	rep(i,1,n) if (deg[i]&1) out("No");
	rep(i,1,n) if (deg[i]>=6) out("Yes");
	int cnt=0;vector<int>v;
	rep(i,1,n) if (deg[i]==4) ++cnt,v.push_back(i);
	if (cnt>=3) out("Yes");
	if (cnt<=1) out("No");
	p1=v[0],p2=v[1];
	dfs(p1,1);
	out((e==2)?"Yes":"No"); 
	return 0;
}