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 <iostream>
#include <algorithm>
#include <string>
#include <vector>
#include <queue>
#include <set>
#include <map>
#include <functional>
#define REP(i, a) for (int i = 0; i < (a); i++)
#define REPR(i, a) for (int i = (int)(a) - 1; i >= 0; i--)
#define FOR(i, a, b) for (int i = (a); i < (b); i++)
#define FORR(i, a, b) for (int i = (int)(b) - 1; i >= a; i--)
#define ALL(a) a.begin(), a.end()
using namespace std;
using ll = long long;
using vi = vector<int>;
using vvi = vector<vi>;
using vl = vector<long long>;
using vvl = vector<vl>;
using vs = vector<string>;
int si() { int x; scanf("%d", &x); return x; }
long long sl() { long long x; scanf("%lld", &x); return x; }
string ss() { string x; cin >> x; return x; }
void pi(int x) { printf("%d ", x); }
void pl(long long x) { printf("%lld ", x); }
void pd(double x) { printf("%.15f ", x); }
void ps(const string &a) { cout << a << ' '; }
void br() { putchar('\n'); }

struct BiconnectedComponent {
  vector<int> ord, low;
  vector<bool> used;
  vector<vector<int>> g;
  vector<vector<pair<int, int>>> bc;
  vector<pair<int, int>> tmp;
  int n, k = 0;
  BiconnectedComponent(const vector<vector<int>> &g) : g(g) {
    n = g.size();
    ord.resize(n, -1);
    low.resize(n, -1);
    used.resize(n, false);
  }
  void dfs(int u, int prev) {
    used[u] = true;
    ord[u] = k ++;
    low[u] = ord[u];
    int cnt = 0;
    for (auto v : g[u]) if (v != prev) {
      if (ord[v] < ord[u]) { 
        tmp.emplace_back(min(u, v), max(u, v));
      }
      if (!used[v]) {
        cnt ++;
        dfs(v, u);
        low[u] = min(low[u], low[v]);
        if (low[v] >= ord[u]) {
          bc.push_back({});
          while (true) {
            pair<int, int> e = tmp.back();
            bc.back().emplace_back(e);
            tmp.pop_back();
            if (min(u, v) == e.first && max(u, v) == e.second) {
              break;
            }
          }
        }
      } else {
        low[u] = min(low[u], ord[v]);
      }
    }
  }
};

bool solve() {
  int N = si(), M = si();
  vvi G(N);
  vector<int> deg(N);
  REP(i, M) {
    int u = si() - 1, v = si() - 1;
    G[u].push_back(v);
    G[v].push_back(u);
    deg[u]++;
    deg[v]++;
  }
  REP(i, N) if (deg[i] % 2 == 1) return false;
  REP(i, N) if (deg[i] >= 6) return true;
  BiconnectedComponent bc(G);
  bc.dfs(0, -1);
  if (bc.bc.size() >= 3) return true;
  if (bc.bc.size() == 1) {
    REP(i, N) if (deg[i] >= 4) return false;
    return true;
  }
  int c = 0;
  REP(i, N) if (deg[i] >= 4) c++;
  return c >= 2;
}

int main() {
  cout << (solve() ? "Yes" : "No") << '\n';
}