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 Rust (1.15.1) (judge verdict: Wrong Answer)

macro_rules! input {
    (source = $s:expr, $($r:tt)*) => {
        let mut iter = $s.split_whitespace();
        let mut next = || { iter.next().unwrap() };
        input_inner!{next, $($r)*}
    };
    ($($r:tt)*) => {
        let stdin = std::io::stdin();
        let mut bytes = std::io::Read::bytes(std::io::BufReader::new(stdin.lock()));
        let mut next = move || -> String {
            bytes
                .by_ref()
                .map(|r| r.unwrap() as char)
                .skip_while(|c| c.is_whitespace())
                .take_while(|c| !c.is_whitespace())
                .collect()
        };
        input_inner!{next, $($r)*}
    };
}

macro_rules! input_inner {
    ($next:expr) => {};
    ($next:expr, ) => {};

    ($next:expr, $var:ident : $t:tt $($r:tt)*) => {
        let $var = read_value!($next, $t);
        input_inner!{$next $($r)*}
    };

    ($next:expr, mut $var:ident : $t:tt $($r:tt)*) => {
        let mut $var = read_value!($next, $t);
        input_inner!{$next $($r)*}
    };
}

macro_rules! read_value {
    ($next:expr, ( $($t:tt),* )) => {
        ( $(read_value!($next, $t)),* )
    };

    ($next:expr, [ $t:tt ; $len:expr ]) => {
        (0..$len).map(|_| read_value!($next, $t)).collect::<Vec<_>>()
    };

    ($next:expr, [ $t:tt ]) => {
        {
            let len = read_value!($next, usize);
            (0..len).map(|_| read_value!($next, $t)).collect::<Vec<_>>()
        }
    };

    ($next:expr, chars) => {
        read_value!($next, String).chars().collect::<Vec<char>>()
    };

    ($next:expr, usize1) => {
        read_value!($next, usize) - 1
    };

    ($next:expr, $t:ty) => {
        $next().parse::<$t>().expect("Parse error")
    };
}

use std::collections::HashSet;

fn main() {
    input!{
        n: usize,
        m: usize,
        v: [(usize1, usize1); m]
    }
    let mut g: Vec<HashSet<usize>> = vec![HashSet::new(); n];
    for &(a, b) in v.iter() {
        g[a].insert(b);
        g[b].insert(a);
    }
    let mut d4: Vec<usize> = Vec::new();
    for i in 0..n {
        let d = g[i].len();
        if d % 2 != 0 {
            println!("No");
            return;
        } else if d >= 6 {
            println!("Yes");
            return;
        } else if d == 4 {
            d4.push(i);
        }
    }
    if d4.len() > 2 {
        println!("Yes");
        return;
    } else if d4.len() < 2 {
        println!("No");
        return;
    }
    let ans = flow(&g, d4[0], d4[1]);
    println!("{}", if ans == 2 {"Yes"} else {"No"});
}

fn flow(g: &Vec<HashSet<usize>>, s: usize, t: usize) -> usize {
    let mut h = g.clone();
    let mut ret = 0;
    loop {
        let mut p: Vec<usize> = Vec::new();
        if dfs(&h, s, t, &mut HashSet::new(), &mut p) {
            for w in p.windows(2) {
                h[w[0]].remove(&w[1]);
                h[w[1]].remove(&w[0]);
            }
            ret += 1;
        } else {
            break;
        }
    }
    ret
}

fn dfs(g: &Vec<HashSet<usize>>, s: usize, t: usize, ps: &mut HashSet<usize>, p: &mut Vec<usize>) -> bool {
    ps.insert(s);
    p.push(s);
    if s == t {
        return true;
    }
    for &c in g[s].iter() {
        if ps.contains(&c) {
            continue;
        }
        if dfs(g, c, t, ps, p) {
            return true;
        }
    }
    ps.remove(&s);
    p.pop();
    return false;
}