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 C++14 (GCC 5.4.1) (judge verdict: Accepted)


#include "bits/stdc++.h"
using namespace std;

#define DEBUG(x) cout<<#x<<": "<<x<<endl;
#define DEBUG_VEC(v) cout<<#v<<":";for(int i=0;i<v.size();i++) cout<<" "<<v[i]; cout<<endl

typedef long long ll;
#define vi vector<int>
#define vl vector<ll>
#define vii vector< vector<int> >
#define vll vector< vector<ll> >
#define vs vector<string>
#define pii pair<int,int>
#define pis pair<int,string>
#define psi pair<string,int>
#define pll pair<ll,ll>
const int inf = 1000000001;
const ll INF = 2e18 * 2;
#define MOD 1000000007
#define mod 1000000009
#define pi 3.14159265358979323846
#define Sp(p) cout<<setprecision(15)<< fixed<<p<<endl;
int dx[4] = { 1,0,-1,0 }, dy[4] = { 0,1,0,-1 };
int dx2[8] = { 1,1,0,-1,-1,-1,0,1 }, dy2[8] = { 0,1,1,1,0,-1,-1,-1 };


#define N 100001
vi par(N); //親
vi ran(N); //木の深さ
vl num(N); //木の要素数

		   //n要素で初期化
void init(int n) {
	int i;
	for (i = 0; i < n; i++) {
		par[i] = i;
		ran[i] = 0;
		num[i] = 1;
	}
}

//木の根を求める
int find(int x) {
	if (par[x] == x) {
		return x;
	}
	else {
		return par[x] = find(par[x]);
	}
}

//xとyの属する集合を併合
void unite(int x, int y) {
	x = find(x);
	y = find(y);
	int nums = num[x] + num[y];
	if (x == y) {
		return;
	}
	if (ran[x]<ran[y]) {
		par[x] = y;
	}
	else {
		par[y] = x;
		if (ran[x] == ran[y]) {
			ran[x]++;
		}
	}
	num[find(x)] = num[find(y)] = nums;
}

//xとyが同じ集合に属するか否か
bool same(int x, int y) {
	return find(x) == find(y);
}



int main() {
	int n, m, i, j;
	cin >> n >> m;
	init(n);
	for (i = 0; i < m; i++) {
		int a, b;
		cin >> a >> b;
		a--; b--;
		unite(a, b);
	}
	ll n1 = num[find(0)], n2 = num[find(1)];
	if (n1 < n2) {
		swap(n1, n2);
	}
	ll ans = (n - n2)*(n - n2 - 1) / 2 + n2*(n2 - 1) / 2 - m;
	cout << ans << endl;
}