#ifndef EVAL
#include "grader.h"
#endif
#include <bits/stdc++.h>
using namespace std;
#define pb push_back
const int MAXN = 1e6 + 5;
vector<int> gr[MAXN];
int deg[MAXN], neig3[MAXN];
int p[MAXN], sz[MAXN];
int n, numcc, total, curcycle;
int great_node;
map<int, int> cnt;
set<int> three;
bool visit[MAXN];
void Init(int _n) {
n = _n;
for (int i = 0; i < n; i++) {
deg[i] = 0;
p[i] = -1;
sz[i] = 1;
gr[i].clear();
}
numcc = n;
total = n;
cnt.clear();
cnt[0] = n;
}
int fin(int x) {
return (p[x] == -1 ? x : p[x] = fin(p[x]));
}
inline void chval(int x, int add) {
if (deg[x] == 3)
three.erase(x);
cnt[deg[x]]--;
if (cnt[deg[x]] == 0)
cnt.erase(deg[x]);
deg[x] += add;
cnt[deg[x]]++;
if (deg[x] == 3)
three.insert(x);
}
void dfs(int u) {
visit[u] = true;
for (auto v : gr[u]) {
if (visit[v]) continue;
dfs(v);
}
}
bool check(int w) {
for (auto v : gr[w])
chval(v, -1);
memset(visit, false, sizeof visit);
visit[w] = true;
int comps = cnt[0] + cnt[1] / 2;
int want = 0;
for (int u = 0; u < n; u++) {
if (!visit[u]) {
want++;
dfs(u);
}
}
for (auto v : gr[w])
chval(v, +1);
return want == comps;
}
void Link(int a, int b) {
if (deg[a] == 3) {
for (auto v : gr[a])
neig3[v]--;
}
if (deg[b] == 3) {
for (auto v : gr[b])
neig3[v]--;
}
chval(a, +1);
chval(b, +1);
gr[a].pb(b);
gr[b].pb(a);
if (deg[a] == 3) {
for (auto v : gr[a])
neig3[v]++;
}
if (deg[b] == 3) {
for (auto v : gr[b])
neig3[v]++;
}
if (deg[a] > 3) {
great_node = a;
} else if (deg[b] > 3) {
great_node = b;
}
int ra = fin(a), rb = fin(b);
if (ra != rb) {
numcc--;
if (sz[ra] > sz[rb]) {
p[rb] = ra;
sz[ra] += sz[rb];
} else {
p[ra] = rb;
sz[rb] += sz[ra];
}
} else {
curcycle = sz[ra];
if (!three.empty() && fin(*three.begin()) != ra) {
total = 0;
return;
}
}
int big = 0;
for (auto it = cnt.rbegin(); it != cnt.rend(); it++) {
if (it->first <= 3) break;
big += it->second;
if (big > 1) break;
}
if (big == 0 && cnt[3] == 0) {
// everything is either a chain or a cycle
int comps = cnt[0] + cnt[1] / 2;
if (comps == numcc) {
// every component is a chain
total = n;
} else if (comps == numcc - 1) {
total = curcycle;
} else {
total = 0;
}
} else if (big > 1 || cnt[3] > 4) {
total = 0;
} else if (big == 1) {
if (neig3[great_node] != (int)three.size()) {
total = 0;
} else {
bool found = (great_node == a || great_node == b);
for (auto v : gr[a])
if (v == great_node)
found = true;
for (auto v : gr[b])
if (v == great_node)
found = true;
if (found) {
if (check(great_node)) {
total = 1;
} else {
total = 0;
}
}
}
} else {
// now big = 0 and cnt[3] > 0
const int c = (int)three.size();
set<int> proc;
for (auto u : three) {
proc.insert(u);
for (auto v : gr[u])
proc.insert(v);
}
bool found = false;
for (auto it : proc) {
if (it == a || it == b)
found = true;
}
if (!found) return;
total = 0;
for (auto u : proc) {
if (deg[u] == 3 && neig3[u] == c - 1) {
if (check(u))
total++;
} else if (deg[u] < 3 && neig3[u] == c) {
if (check(u))
total++;
}
}
}
}
int CountCritical() {
return total;
}
