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



Sample input:
7 9
1 2
1 3
2 3
1 4
1 5
4 5
1 6
1 7
6 7


Sample output:
Yes

Submission status: Runtime Error

Solution code:
n, m = map(int, input().split())
edge = [tuple(map(int, input().split())) for _ in range(m)]
connect = [set() for _ in range(n)]
for a, b in edge:
    a -= 1
    b -= 1
    connect[a].add(b)
    connect[b].add(a)
def search(cur, start):
    if start in connect[cur]:
        connect[cur].discard(start)
        connect[start].discard(cur)
    elif connect[cur] == set():
        print('No')
        exit()
    else:
        x = connect[cur].pop()
        connect[x].discard(cur)
        search(x, start)
for _ in range(3):
    # check not all set
    allset = True
    for s in connect:
        if s != set():
            allset = False
    if allset:
        print('No')
        exit()
    for a, b in edge:
        a -= 1
        b -= 1
        if a in connect[b]:
            connect[a].discard(b)
            connect[b].discard(a)
            search(a, b)
            break
print('Yes')

Submission status: Runtime Error

Solution code:
n,m = map(int,input().split())
adj = []
for i in range(m):
  adj.append(list(map(int,input().split())))
assert(adj[0][0] != 1)  
if m > n+2:
  print("Yes")
elif m <= n+2:
  print("No")
else:
  assert(1==0)

Submission status: Runtime Error

Solution code:
n,m = map(int,input().split())
adj = []
for i in range(m):
  adj.append(list(map(int,input().split())))
assert(adj[0][0] == 1)  
if m > n+2:
  print("Yes")
elif m <= n+2:
  print("No")
else:
  assert(1==0)

Submission status: Runtime Error

Solution code:
loop_count = 0

def dfs(graph, path):
    global loop_count
    a = path[-1]
    nexts = list(graph[path[-1]])
    for nex in nexts:
        try:
            graph[a].remove(nex)
            graph[nex].remove(a)
        except KeyError:
            continue
        if nex in path:
            loop_count += 1
            path.pop()
            return nex
        else:
            path.append(nex)
            ret = dfs(graph, path)
            if ret is None:
                return
            if ret != a:
                path.pop()
                return ret


def main():
    n, m = map(int, raw_input().split())
    graph = {}
    for _ in xrange(m):
        a, b = map(int, raw_input().split())
        if a not in graph:
            graph[a] = set()
        if b not in graph:
            graph[b] = set()
        graph[a].add(b)
        graph[b].add(a)
    dfs(graph, [1])
    if loop_count < 3 or any(len(g) > 0 for g in graph.itervalues()):
        print 'No'
    else:
        print 'Yes'


if __name__ == "__main__":
    main()

Submission status: Runtime Error

Solution code:
n,m = map(int,input().split())
if m > n+2:
  #print("Yes")
  assert(1 == 0)
elif m < n+2:
  print("No")
else:
  assert(1 ==0)

Submission status: Runtime Error

Solution code:
n,m = map(int,input().split())
if m > n+2:
  print("Yes")
elif m < n+2:
  print("No")
else:
  assert(1 ==0)

Submission status: Runtime Error

Solution code:
n = int(input())
list = []
a = []
for i in range(n):
  list.append(int(input))

for i in range(1, n):
  for j in range(1, i):
    if list[i] == j:
      a.insert(j, j)

if a == list:
  for i in range(n):
    print(list[i])
else:
  print(-1)

Submission status: Runtime Error

Solution code:
n,m = map(int,input().split())
if m < n+2:
  print("No")
else:
  assert(1==0)

Submission status: Runtime Error

Solution code:
n,m = map(int,input().split())
if m >= n+2:
  print("Yes")
else:
  assert(1==0)

Submission status: Runtime Error

Solution code:
#!/usr/bin/python
# -*- coding: utf-8 -*-
import sys
import math
import collections


N, M = map(int, sys.stdin.readline().rstrip().split(" "))

pathes = collections.defaultdict(list)
for _ in range(M):
    a, b = map(int, sys.stdin.readline().rstrip().split(" "))
    pathes[a].append(b)
    pathes[b].append(a)


def find_root():
    for key, value in pathes.items():
        if len(value) > 2:
            return key
    for key, value in pathes.items():
        if len(value) == 2:
            return key

def traverse(root, value):
    if len(pathes[value]) == 0:
        return False
    next_value = pathes[value][0]
    del(pathes[value][0])

    for i in range(len(pathes[next_value])):
        if pathes[next_value][i] == value:
            del(pathes[next_value][i])
            break
    if root == next_value:
        return True
    else:
        return traverse(root, next_value)

count = 0
while True:
    # print(pathes)
    root = find_root()
    if root is None:
        break
    # print(root)
    is_valid = traverse(root, root)
    if not is_valid:
        print("No")
        exit(0)

    count += 1

rest = sum(len(counts) for counts in pathes.values())
if rest > 0:
    print("No")
else:
    if count >=3:
        print("Yes")
    else:
        print("No")

exit(0)

Submission status: Runtime Error

Solution code:
#!/usr/bin/python
# -*- coding: utf-8 -*-
import sys
import math
import collections


N, M = map(int, sys.stdin.readline().rstrip().split(" "))

pathes = collections.defaultdict(list)
for _ in range(M):
    a, b = map(int, sys.stdin.readline().rstrip().split(" "))
    pathes[a].append(b)
    pathes[b].append(a)


def find_root():
    for key, value in pathes.items():
        if len(value) > 2:
            return key
    for key, value in pathes.items():
        if len(value) == 2:
            return key

def traverse(root, value):
    if len(pathes[value]) == 0:
        return False
    next_value = pathes[value][0]
    del(pathes[value][0])

    for i in range(len(pathes[next_value])):
        if pathes[next_value][i] == value:
            del(pathes[next_value][i])
            break
    if root == next_value:
        return True
    else:
        traverse(root, next_value)

count = 0
while True:
    # print(pathes)
    root = find_root()
    if root is None:
        break
    # print(root)
    is_valid = traverse(root, root)
    if not is_valid:
        print("No")
        exit(0)

    count += 1

rest = sum(len(counts) for counts in pathes.values())
if rest > 0:
    print("No")
else:
    if count >=3:
        print("Yes")
    else:
        print("No")

exit(0)

Submission status: Runtime Error

Solution code:
#!/usr/bin/python
# -*- coding: utf-8 -*-
import sys
import math
import collections


N, M = map(int, sys.stdin.readline().rstrip().split(" "))

pathes = collections.defaultdict(list)
for _ in range(M):
    a, b = map(int, sys.stdin.readline().rstrip().split(" "))
    pathes[a].append(b)
    pathes[b].append(a)


def find_root():
    for key, value in pathes.items():
        if len(value) > 2:
            return key
    for key, value in pathes.items():
        if len(value) == 2:
            return key

def traverse(root, value):
    next_value = pathes[value][0]
    del(pathes[value][0])

    for i in range(len(pathes[next_value])):
        if pathes[next_value][i] == value:
            del(pathes[next_value][i])
            break
    if root == next_value:
        return
    else:
        traverse(root, next_value)

count = 0
while True:
    # print(pathes)
    root = find_root()
    if root is None:
        break
    # print(root)
    traverse(root, root)
    count += 1

rest = sum(len(counts) for counts in pathes.values())
if rest > 0:
    print("No")
else:
    if count >=3:
        print("Yes")
    else:
        print("No")

exit(0)

Submission status: Runtime Error

Solution code:
#include <stdio.h>
#include <vector>
#include <set>
#include <deque>

int t1,t2;
int n,m;

int cnt[100005];
std::set<int> g[100005];


std::deque<int> d1,d2;
void dfs(int u)
{
    d1.push_back(u);
    while (!g[u].empty()) {
        int v=*g[u].begin();
        g[u].erase(v);
        g[v].erase(u);
        dfs(v);
    }
    d2.push_front(d1.back());
    d1.pop_back();
}

int has[100005];
std::vector<int> comps;


int main()
{
    #ifdef NOT_DMOJ
    freopen("data.txt","r",stdin);
    #endif // NOT_DMOJ
    scanf("%d%d",&n,&m);
    for (int i=0; i<m; i++) {
        scanf("%d%d",&t1,&t2);
        g[t1].insert(t2);
        g[t2].insert(t1);
        cnt[t1]++;
        cnt[t2]++;
    }
    for (int i=1; i<=n; i++) {
        if (cnt[i]&1) {
            printf("No\n");
            return 0;
        }
    }
    for (int i=1; i<=n; i++) {
        if (g[i].empty()) continue;
        d1.clear();
        d2.clear();
        dfs(i);
        //for (int i:d2) printf("%d ",i); printf("\n");
        // count number of cycles
        t1=0;
        while (!d2.empty()) {
            if (has[d2.front()]) {
                // pop cycle
                //printf("popping\n");
                t1++;
                while (has[d2.front()]) {
                    has[d1.back()]=0;
                    d1.pop_back();
                }
            }
            has[d2.front()]=1;
            d1.push_back(d2.front());
            d2.pop_front();
        }
        comps.push_back(t1);
    }
    if (comps.size()>3) printf("No\n");
    else {
        int sum=0;
        for (int i:comps) sum+=i;
        if (sum<3) printf("No\n");
        else printf("Yes\n");
    }
}