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)

// たなこふ先生の input マクロ (´･_･`)
// https://qiita.com/tanakh/items/0ba42c7ca36cd29d0ac8

#[allow(unused_macros)]
macro_rules! input {
    (source = $s:expr, $($r:tt)*) => {
        let mut iter = $s.split_whitespace();
        input_inner!{iter, $($r)*}
    };
    ($($r:tt)*) => {
        let s = {
            use std::io::Read;
            let mut s = String::new();
            std::io::stdin().read_to_string(&mut s).unwrap();
            s
        };
        let mut iter = s.split_whitespace();
        input_inner!{iter, $($r)*}
    };
}

#[allow(unused_macros)]
macro_rules! input_inner {
    ($iter:expr) => {};
    ($iter:expr, ) => {};

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

#[allow(unused_macros)]
macro_rules! read_value {
    ($iter:expr, ( $($t:tt),* )) => {
        ( $(read_value!($iter, $t)),* )
    };

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

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

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

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

struct UnionFind {
    par: Vec<usize>,
}

impl UnionFind {
    fn new(n: usize) -> UnionFind {
        UnionFind {
            par: (0..n).collect(),
        }
    }

    fn root(&mut self, x: usize) -> usize {
        let p = self.par[x];
        if p == x {
            x
        } else {
            let r = self.root(p);
            self.par[x] = r;
            r
        }
    }

    fn same(&mut self, x: usize, y: usize) -> bool {
        self.root(x) == self.root(y)
    }

    fn unite(&mut self, x: usize, y: usize) {
        let rootx = self.root(x);
        let rooty = self.root(y);
        if rootx == rooty {
            ;
        } else {
            self.par[rootx] = rooty;
        }
    }
}

type Graph = (Vec<Vec<usize>>, Vec<(usize, usize)>);
use std::collections::HashSet;

fn solv((nodes, edges): Graph) -> bool {
    let mut uf = UnionFind::new(edges.len());
    let mut deg4nodes = HashSet::new();

    for i in 0..edges.len() {
        let (a, b) = edges[i];
        if nodes[a].len() % 2 == 1 { return false; }
        if nodes[b].len() % 2 == 1 { return false; }
        if nodes[a].len() > 4 { return true; }
        if nodes[b].len() > 4 { return true; }

        if nodes[a].len() > 2 {
            deg4nodes.insert(a);
        } else {
            uf.unite(nodes[a][0], nodes[a][1]);
        }
        if nodes[b].len() > 2 {
            deg4nodes.insert(b);
        } else {
            uf.unite(nodes[b][0], nodes[b][1]);
        }
    }

    if deg4nodes.len() > 2 { return true; }
    if deg4nodes.len() < 2 { return false; }

    let &n = deg4nodes.iter().nth(0).unwrap();

    return
        uf.same(nodes[n][0], nodes[n][1]) ||
        uf.same(nodes[n][0], nodes[n][2]) ||
        uf.same(nodes[n][0], nodes[n][3]) ||
        uf.same(nodes[n][1], nodes[n][2]) ||
        uf.same(nodes[n][1], nodes[n][3]) ||
        uf.same(nodes[n][2], nodes[n][3]);
}

fn main() {
    input! {
        n: usize,
        m: usize,
        edges: [(usize1, usize1); m],
    }

    let mut nodes = vec![vec![]; n];

    for i in 0..edges.len() {
        let (a, b) = edges[i];
        nodes[a].push(i);
        nodes[b].push(i);
    }

    if solv((nodes, edges)) {
        println!("Yes");
    } else {
        println!("No");
    }
}