Problem:
Score : 100 points

Problem Statement

You are given an undirected graph G. G has N vertices and M edges. The vertices are numbered from 1 through N, and the i-th edge (1 ≤ i ≤ M) connects Vertex a_i and b_i. G does not have self-loops and multiple edges.

You can repeatedly perform the operation of adding an edge between two vertices. However, G must not have self-loops or multiple edges as the result. Also, if Vertex 1 and 2 are connected directly or indirectly by edges, your body will be exposed to a voltage of 1000000007 volts. This must also be avoided.

Under these conditions, at most how many edges can you add? Note that Vertex 1 and 2 are never connected directly or indirectly in the beginning.

Constraints

2 ≤ N ≤ 10^5

0 ≤ M ≤ 10^5

1 ≤ a_i < b_i ≤ N

All pairs (a_i, b_i) are distinct.

Vertex 1 and 2 in G are not connected directly or indirectly by edges.

Input

Input is given from Standard Input in the following format:

N M
a_1 b_1
:
a_M b_M

Output

Print the maximum number of edges that can be added.

Sample Input 1

4 1
1 3

Sample Output 1

2

As shown above, two edges can be added. It is not possible to add three or more edges.

Sample Input 2

2 0

Sample Output 2

0

No edge can be added.

Sample Input 3

9 6
1 4
1 8
2 5
3 6
4 8
5 7

Sample Output 3

12

Solution in Rust (1.15.1) (judge verdict: Accepted)

use std::io::*;

mod disjoint_set {
    pub struct DisjointSet {
        parent: Vec<usize>,
        size: Vec<usize>,
    }
    impl DisjointSet {
        pub fn new(n: usize) -> DisjointSet {
            DisjointSet {
                parent: (0..n).collect(),
                size: vec![1; n],
            }
        }
        fn root(&mut self, x: usize) -> usize {
            if x == self.parent[x] {
                x
            } else {
                let tmp = self.parent[x];
                self.parent[x] = self.root(tmp);
                self.parent[x]
            }
        }
        pub fn merge(&mut self, x: usize, y: usize) {
            let x = self.root(x);
            let y = self.root(y);
            if x == y {
                return;
            }
            if self.size[x] > self.size[y] {
                self.size[x] += self.size[y];
                self.parent[y] = x;
            } else {
                self.size[y] += self.size[x];
                self.parent[x] = y;
            }
        }
        pub fn same(&mut self, x: usize, y: usize) -> bool {
            self.root(x) == self.root(y)
        }
        pub fn size(&mut self, x: usize) -> usize {
            let x = self.root(x);
            self.size[x]
        }
    }
}

fn main() {
    let mut s: String = String::new();
    std::io::stdin().read_to_string(&mut s).ok();
    let mut itr = s.trim().split_whitespace();
    let n: usize = itr.next().unwrap().parse().unwrap();
    let m: usize = itr.next().unwrap().parse().unwrap();
    let a: Vec<(usize, usize)> = (0..m)
        .map(|_| {
            (
                itr.next().unwrap().parse().unwrap(),
                itr.next().unwrap().parse().unwrap(),
            )
        })
        .collect();

    let mut uf = disjoint_set::DisjointSet::new(n + 1);
    for i in 0..m {
        uf.merge(a[i].0, a[i].1);
    }
    let mut one_vert = uf.size(1);
    let mut two_vert = uf.size(2);
    let other_vert = n - one_vert - two_vert;
    let mut ans = 0;
    if one_vert < two_vert {
        std::mem::swap(&mut one_vert, &mut two_vert);
    }
    // 完全グラフの辺の数 - 今の辺
    ans += (one_vert + other_vert) * (one_vert + other_vert - 1) / 2;
    ans += two_vert * (two_vert - 1) / 2;
    println!("{}", ans - m);
}