Buggy python submission (verdict: Time Limit Exceeded):
#!/usr/bin/env python3
# vim: set fileencoding=utf-8

# pylint: disable=unused-import, invalid-name, missing-docstring, bad-continuation


"""Module docstring
"""

import functools
import heapq
import itertools
import logging
import math
import os
import random
import string
import sys
from argparse import ArgumentParser
from collections import defaultdict, deque
from copy import deepcopy
from io import BytesIO, IOBase
from typing import Dict, List, Optional, Set, Tuple


def solve(values: List[Tuple[int, int]], n: int) -> List[int]:
    passages = defaultdict(set)
    for a, b in values:
        passages[a].add(b)
        passages[b].add(a)
    if len(passages) < n:
        return []
    to_visit = deque((1,))
    visited = set()
    # 0 idx is not used
    distances = [n + 1] * (n + 1)
    distances[1] = 0
    signs = [1] * (n + 1)
    while to_visit:
        cur = to_visit.popleft()
        for neighbour in passages[cur]:
            tmp = distances[cur] + 1
            if tmp < distances[neighbour]:
                distances[neighbour] = tmp
                signs[neighbour] = cur
            if neighbour not in visited:
                to_visit.append(neighbour)
        visited.add(cur)
    # LOG.debug((visited))
    # LOG.debug((distances))
    if len(visited) < n:
        return []
    return signs[2:]


def do_job(stdin, stdout):
    "Do the work"
    LOG.debug("Start working")
    N, M = map(int, stdin.readline().split())
    if M < N - 1:
        print("No", file=stdout)
        return
    values = []
    for _ in range(M):
        a, b = map(int, stdin.readline().split())
        values.append((a, b))
    result = solve(values, N)
    if not result:
        print("No", file=stdout)
    else:
        print("Yes", file=stdout)
        print("\n".join(map(str, result)), file=stdout)


def print_output(testcase: int, result, stdout) -> None:
    "Formats and print result"
    if result is None:
        result = "IMPOSSIBLE"
    print("Case #{}: {}".format(testcase + 1, result), file=stdout)
    # 6 digits float precision {:.6f} (6 is the default value)
    # print("Case #{}: {:f}".format(testcase + 1, result), file=stdout)


BUFSIZE = 8192


class FastIO(IOBase):
    # pylint: disable=super-init-not-called, expression-not-assigned
    newlines = 0

    def __init__(self, file):
        self._fd = file.fileno()
        self.buffer = BytesIO()
        self.writable = "x" in file.mode or "r" not in file.mode
        self.write = self.buffer.write if self.writable else None

    def read(self):
        while True:
            b = os.read(self._fd, max(os.fstat(self._fd).st_size, BUFSIZE))
            if not b:
                break
            ptr = self.buffer.tell()
            self.buffer.seek(0, 2), self.buffer.write(b), self.buffer.seek(ptr)
        self.newlines = 0
        return self.buffer.read()

    def readline(self):
        while self.newlines == 0:
            b = os.read(self._fd, max(os.fstat(self._fd).st_size, BUFSIZE))
            self.newlines = b.count(b"\n") + (not b)
            ptr = self.buffer.tell()
            self.buffer.seek(0, 2), self.buffer.write(b), self.buffer.seek(ptr)
        self.newlines -= 1
        return self.buffer.readline()

    def flush(self):
        if self.writable:
            os.write(self._fd, self.buffer.getvalue())
            self.buffer.truncate(0), self.buffer.seek(0)


class IOWrapper(IOBase):
    # pylint: disable=super-init-not-called
    def __init__(self, file):
        self.buffer = FastIO(file)
        self.flush = self.buffer.flush
        self.writable = self.buffer.writable
        self.write = lambda s: self.buffer.write(s.encode("ascii"))
        self.read = lambda: self.buffer.read().decode("ascii")
        self.readline = lambda: self.buffer.readline().decode("ascii")


def configure_log() -> None:
    "Configure the log output"
    log_formatter = logging.Formatter("L%(lineno)d - " "%(message)s")
    handler = logging.StreamHandler(IOWrapper(sys.stderr))
    handler.setFormatter(log_formatter)
    LOG.addHandler(handler)


LOG = None
# for interactive call: do not add multiple times the handler
if not LOG:
    LOG = logging.getLogger("template")
    configure_log()


def main(argv=None):
    "Program wrapper."
    if argv is None:
        argv = sys.argv[1:]
    parser = ArgumentParser()
    parser.add_argument(
        "-v",
        "--verbose",
        dest="verbose",
        action="store_true",
        default=False,
        help="run as verbose mode",
    )
    args = parser.parse_args(argv)
    if args.verbose:
        LOG.setLevel(logging.DEBUG)
    stdin = IOWrapper(sys.stdin)
    stdout = IOWrapper(sys.stdout)
    do_job(stdin, stdout)
    stdout.flush()
    for h in LOG.handlers:
        h.flush()
    return 0


if __name__ == "__main__":
    sys.exit(main())


Fixed submission:
#!/usr/bin/env python3
# vim: set fileencoding=utf-8

# pylint: disable=unused-import, invalid-name, missing-docstring, bad-continuation


"""Module docstring
"""

import functools
import heapq
import itertools
import logging
import math
import os
import random
import string
import sys
from argparse import ArgumentParser
from collections import defaultdict, deque
from copy import deepcopy
from io import BytesIO, IOBase
from typing import Dict, List, Optional, Set, Tuple


def solve(values: List[Tuple[int, int]], n: int) -> List[int]:
    passages = defaultdict(set)
    for a, b in values:
        passages[a].add(b)
        passages[b].add(a)
    if len(passages) < n:
        return []
    to_visit = [(0, 1, 0)]
    visited = set()
    # 0 idx is not used
    signs = [1] * (n + 1)
    while to_visit:
        min_dist, cur, prev = heapq.heappop(to_visit)
        if cur in visited:
            continue
        signs[cur] = prev
        for neighbour in passages[cur]:
            if neighbour in visited:
                continue
            heapq.heappush(to_visit, (min_dist + 1, neighbour, cur))
        visited.add(cur)
    LOG.debug((visited))
    LOG.debug((signs))
    if len(visited) < n:
        return []
    return signs[2:]


def do_job(stdin, stdout):
    "Do the work"
    LOG.debug("Start working")
    N, M = map(int, stdin.readline().split())
    if M < N - 1:
        print("No", file=stdout)
        return
    values = []
    for _ in range(M):
        a, b = map(int, stdin.readline().split())
        values.append((a, b))
    result = solve(values, N)
    if not result:
        print("No", file=stdout)
    else:
        print("Yes", file=stdout)
        print("\n".join(map(str, result)), file=stdout)


def print_output(testcase: int, result, stdout) -> None:
    "Formats and print result"
    if result is None:
        result = "IMPOSSIBLE"
    print("Case #{}: {}".format(testcase + 1, result), file=stdout)
    # 6 digits float precision {:.6f} (6 is the default value)
    # print("Case #{}: {:f}".format(testcase + 1, result), file=stdout)


BUFSIZE = 8192


class FastIO(IOBase):
    # pylint: disable=super-init-not-called, expression-not-assigned
    newlines = 0

    def __init__(self, file):
        self._fd = file.fileno()
        self.buffer = BytesIO()
        self.writable = "x" in file.mode or "r" not in file.mode
        self.write = self.buffer.write if self.writable else None

    def read(self):
        while True:
            b = os.read(self._fd, max(os.fstat(self._fd).st_size, BUFSIZE))
            if not b:
                break
            ptr = self.buffer.tell()
            self.buffer.seek(0, 2), self.buffer.write(b), self.buffer.seek(ptr)
        self.newlines = 0
        return self.buffer.read()

    def readline(self):
        while self.newlines == 0:
            b = os.read(self._fd, max(os.fstat(self._fd).st_size, BUFSIZE))
            self.newlines = b.count(b"\n") + (not b)
            ptr = self.buffer.tell()
            self.buffer.seek(0, 2), self.buffer.write(b), self.buffer.seek(ptr)
        self.newlines -= 1
        return self.buffer.readline()

    def flush(self):
        if self.writable:
            os.write(self._fd, self.buffer.getvalue())
            self.buffer.truncate(0), self.buffer.seek(0)


class IOWrapper(IOBase):
    # pylint: disable=super-init-not-called
    def __init__(self, file):
        self.buffer = FastIO(file)
        self.flush = self.buffer.flush
        self.writable = self.buffer.writable
        self.write = lambda s: self.buffer.write(s.encode("ascii"))
        self.read = lambda: self.buffer.read().decode("ascii")
        self.readline = lambda: self.buffer.readline().decode("ascii")


def configure_log() -> None:
    "Configure the log output"
    log_formatter = logging.Formatter("L%(lineno)d - " "%(message)s")
    handler = logging.StreamHandler(IOWrapper(sys.stderr))
    handler.setFormatter(log_formatter)
    LOG.addHandler(handler)


LOG = None
# for interactive call: do not add multiple times the handler
if not LOG:
    LOG = logging.getLogger("template")
    configure_log()


def main(argv=None):
    "Program wrapper."
    if argv is None:
        argv = sys.argv[1:]
    parser = ArgumentParser()
    parser.add_argument(
        "-v",
        "--verbose",
        dest="verbose",
        action="store_true",
        default=False,
        help="run as verbose mode",
    )
    args = parser.parse_args(argv)
    if args.verbose:
        LOG.setLevel(logging.DEBUG)
    stdin = IOWrapper(sys.stdin)
    stdout = IOWrapper(sys.stdout)
    do_job(stdin, stdout)
    stdout.flush()
    for h in LOG.handlers:
        h.flush()
    return 0


if __name__ == "__main__":
    sys.exit(main())

Buggy python submission (verdict: Time Limit Exceeded):
#!/usr/bin/env python3
# vim: set fileencoding=utf-8

# pylint: disable=unused-import, invalid-name, missing-docstring, bad-continuation


"""Module docstring
"""

import functools
import heapq
import itertools
import logging
import math
import os
import random
import string
import sys
from argparse import ArgumentParser
from collections import defaultdict, deque
from copy import deepcopy
from io import BytesIO, IOBase
from typing import Dict, List, Optional, Set, Tuple


def solve(values: List[Tuple[int, int]], n: int) -> List[int]:
    passages = defaultdict(set)
    for a, b in values:
        passages[a].add(b)
        passages[b].add(a)
    if len(passages) < n:
        return []
    to_visit = deque((1,))
    visited = set()
    # 0 idx is not used
    distances = [n + 1] * (n + 1)
    distances[1] = 0
    signs = [1] * (n + 1)
    while to_visit:
        cur = to_visit.popleft()
        for neighbour in passages[cur]:
            tmp = distances[cur] + 1
            if tmp < distances[neighbour]:
                distances[neighbour] = tmp
                signs[neighbour] = cur
            if neighbour not in visited:
                to_visit.append(neighbour)
        visited.add(cur)
    # LOG.debug((visited))
    # LOG.debug((distances))
    if len(visited) < n:
        return []
    return signs[2:]


def do_job(stdin, stdout):
    "Do the work"
    LOG.debug("Start working")
    N, M = map(int, stdin.readline().split())
    values = []
    for _ in range(M):
        a, b = map(int, stdin.readline().split())
        values.append((a, b))
    result = solve(values, N)
    if not result:
        print("No", file=stdout)
    else:
        print("Yes", file=stdout)
        print("\n".join(map(str, result)), file=stdout)


def print_output(testcase: int, result, stdout) -> None:
    "Formats and print result"
    if result is None:
        result = "IMPOSSIBLE"
    print("Case #{}: {}".format(testcase + 1, result), file=stdout)
    # 6 digits float precision {:.6f} (6 is the default value)
    # print("Case #{}: {:f}".format(testcase + 1, result), file=stdout)


BUFSIZE = 8192


class FastIO(IOBase):
    # pylint: disable=super-init-not-called, expression-not-assigned
    newlines = 0

    def __init__(self, file):
        self._fd = file.fileno()
        self.buffer = BytesIO()
        self.writable = "x" in file.mode or "r" not in file.mode
        self.write = self.buffer.write if self.writable else None

    def read(self):
        while True:
            b = os.read(self._fd, max(os.fstat(self._fd).st_size, BUFSIZE))
            if not b:
                break
            ptr = self.buffer.tell()
            self.buffer.seek(0, 2), self.buffer.write(b), self.buffer.seek(ptr)
        self.newlines = 0
        return self.buffer.read()

    def readline(self):
        while self.newlines == 0:
            b = os.read(self._fd, max(os.fstat(self._fd).st_size, BUFSIZE))
            self.newlines = b.count(b"\n") + (not b)
            ptr = self.buffer.tell()
            self.buffer.seek(0, 2), self.buffer.write(b), self.buffer.seek(ptr)
        self.newlines -= 1
        return self.buffer.readline()

    def flush(self):
        if self.writable:
            os.write(self._fd, self.buffer.getvalue())
            self.buffer.truncate(0), self.buffer.seek(0)


class IOWrapper(IOBase):
    # pylint: disable=super-init-not-called
    def __init__(self, file):
        self.buffer = FastIO(file)
        self.flush = self.buffer.flush
        self.writable = self.buffer.writable
        self.write = lambda s: self.buffer.write(s.encode("ascii"))
        self.read = lambda: self.buffer.read().decode("ascii")
        self.readline = lambda: self.buffer.readline().decode("ascii")


def configure_log() -> None:
    "Configure the log output"
    log_formatter = logging.Formatter("L%(lineno)d - " "%(message)s")
    handler = logging.StreamHandler(IOWrapper(sys.stderr))
    handler.setFormatter(log_formatter)
    LOG.addHandler(handler)


LOG = None
# for interactive call: do not add multiple times the handler
if not LOG:
    LOG = logging.getLogger("template")
    configure_log()


def main(argv=None):
    "Program wrapper."
    if argv is None:
        argv = sys.argv[1:]
    parser = ArgumentParser()
    parser.add_argument(
        "-v",
        "--verbose",
        dest="verbose",
        action="store_true",
        default=False,
        help="run as verbose mode",
    )
    args = parser.parse_args(argv)
    if args.verbose:
        LOG.setLevel(logging.DEBUG)
    stdin = IOWrapper(sys.stdin)
    stdout = IOWrapper(sys.stdout)
    do_job(stdin, stdout)
    stdout.flush()
    for h in LOG.handlers:
        h.flush()
    return 0


if __name__ == "__main__":
    sys.exit(main())


Fixed submission:
#!/usr/bin/env python3
# vim: set fileencoding=utf-8

# pylint: disable=unused-import, invalid-name, missing-docstring, bad-continuation


"""Module docstring
"""

import functools
import heapq
import itertools
import logging
import math
import os
import random
import string
import sys
from argparse import ArgumentParser
from collections import defaultdict, deque
from copy import deepcopy
from io import BytesIO, IOBase
from typing import Dict, List, Optional, Set, Tuple


def solve(values: List[Tuple[int, int]], n: int) -> List[int]:
    passages = defaultdict(set)
    for a, b in values:
        passages[a].add(b)
        passages[b].add(a)
    if len(passages) < n:
        return []
    to_visit = [(0, 1, 0)]
    visited = set()
    # 0 idx is not used
    signs = [1] * (n + 1)
    while to_visit:
        min_dist, cur, prev = heapq.heappop(to_visit)
        if cur in visited:
            continue
        signs[cur] = prev
        for neighbour in passages[cur]:
            if neighbour in visited:
                continue
            heapq.heappush(to_visit, (min_dist + 1, neighbour, cur))
        visited.add(cur)
    LOG.debug((visited))
    LOG.debug((signs))
    if len(visited) < n:
        return []
    return signs[2:]


def do_job(stdin, stdout):
    "Do the work"
    LOG.debug("Start working")
    N, M = map(int, stdin.readline().split())
    if M < N - 1:
        print("No", file=stdout)
        return
    values = []
    for _ in range(M):
        a, b = map(int, stdin.readline().split())
        values.append((a, b))
    result = solve(values, N)
    if not result:
        print("No", file=stdout)
    else:
        print("Yes", file=stdout)
        print("\n".join(map(str, result)), file=stdout)


def print_output(testcase: int, result, stdout) -> None:
    "Formats and print result"
    if result is None:
        result = "IMPOSSIBLE"
    print("Case #{}: {}".format(testcase + 1, result), file=stdout)
    # 6 digits float precision {:.6f} (6 is the default value)
    # print("Case #{}: {:f}".format(testcase + 1, result), file=stdout)


BUFSIZE = 8192


class FastIO(IOBase):
    # pylint: disable=super-init-not-called, expression-not-assigned
    newlines = 0

    def __init__(self, file):
        self._fd = file.fileno()
        self.buffer = BytesIO()
        self.writable = "x" in file.mode or "r" not in file.mode
        self.write = self.buffer.write if self.writable else None

    def read(self):
        while True:
            b = os.read(self._fd, max(os.fstat(self._fd).st_size, BUFSIZE))
            if not b:
                break
            ptr = self.buffer.tell()
            self.buffer.seek(0, 2), self.buffer.write(b), self.buffer.seek(ptr)
        self.newlines = 0
        return self.buffer.read()

    def readline(self):
        while self.newlines == 0:
            b = os.read(self._fd, max(os.fstat(self._fd).st_size, BUFSIZE))
            self.newlines = b.count(b"\n") + (not b)
            ptr = self.buffer.tell()
            self.buffer.seek(0, 2), self.buffer.write(b), self.buffer.seek(ptr)
        self.newlines -= 1
        return self.buffer.readline()

    def flush(self):
        if self.writable:
            os.write(self._fd, self.buffer.getvalue())
            self.buffer.truncate(0), self.buffer.seek(0)


class IOWrapper(IOBase):
    # pylint: disable=super-init-not-called
    def __init__(self, file):
        self.buffer = FastIO(file)
        self.flush = self.buffer.flush
        self.writable = self.buffer.writable
        self.write = lambda s: self.buffer.write(s.encode("ascii"))
        self.read = lambda: self.buffer.read().decode("ascii")
        self.readline = lambda: self.buffer.readline().decode("ascii")


def configure_log() -> None:
    "Configure the log output"
    log_formatter = logging.Formatter("L%(lineno)d - " "%(message)s")
    handler = logging.StreamHandler(IOWrapper(sys.stderr))
    handler.setFormatter(log_formatter)
    LOG.addHandler(handler)


LOG = None
# for interactive call: do not add multiple times the handler
if not LOG:
    LOG = logging.getLogger("template")
    configure_log()


def main(argv=None):
    "Program wrapper."
    if argv is None:
        argv = sys.argv[1:]
    parser = ArgumentParser()
    parser.add_argument(
        "-v",
        "--verbose",
        dest="verbose",
        action="store_true",
        default=False,
        help="run as verbose mode",
    )
    args = parser.parse_args(argv)
    if args.verbose:
        LOG.setLevel(logging.DEBUG)
    stdin = IOWrapper(sys.stdin)
    stdout = IOWrapper(sys.stdout)
    do_job(stdin, stdout)
    stdout.flush()
    for h in LOG.handlers:
        h.flush()
    return 0


if __name__ == "__main__":
    sys.exit(main())

Buggy python submission (verdict: Time Limit Exceeded):
#!/usr/bin/env python3
# vim: set fileencoding=utf-8

# pylint: disable=unused-import, invalid-name, missing-docstring, bad-continuation


"""Module docstring
"""

import functools
import heapq
import itertools
import logging
import math
import os
import random
import string
import sys
from argparse import ArgumentParser
from collections import defaultdict, deque
from copy import deepcopy
from io import BytesIO, IOBase
from typing import Dict, List, Optional, Set, Tuple


def solve(values: List[Tuple[int, int]], n: int) -> List[int]:
    passages = defaultdict(set)
    for a, b in values:
        passages[a].add(b)
        passages[b].add(a)
    to_visit = deque((1,))
    visited = set()
    # 0 idx is not used
    distances = [n + 1] * (n + 1)
    distances[1] = 0
    signs = [1] * (n + 1)
    while to_visit:
        cur = to_visit.popleft()
        for neighbour in passages[cur]:
            tmp = distances[cur] + 1
            if tmp < distances[neighbour]:
                distances[neighbour] = tmp
                signs[neighbour] = cur
            if neighbour not in visited:
                to_visit.append(neighbour)
        visited.add(cur)
    # LOG.debug((visited))
    # LOG.debug((distances))
    if len(visited) < n:
        return []
    return signs[2:]


def do_job(stdin, stdout):
    "Do the work"
    LOG.debug("Start working")
    N, M = map(int, stdin.readline().split())
    values = []
    for _ in range(M):
        a, b = map(int, stdin.readline().split())
        values.append((a, b))
    result = solve(values, N)
    if not result:
        print("No", file=stdout)
    else:
        print("Yes", file=stdout)
        print("\n".join(map(str, result)), file=stdout)


def print_output(testcase: int, result, stdout) -> None:
    "Formats and print result"
    if result is None:
        result = "IMPOSSIBLE"
    print("Case #{}: {}".format(testcase + 1, result), file=stdout)
    # 6 digits float precision {:.6f} (6 is the default value)
    # print("Case #{}: {:f}".format(testcase + 1, result), file=stdout)


BUFSIZE = 8192


class FastIO(IOBase):
    # pylint: disable=super-init-not-called, expression-not-assigned
    newlines = 0

    def __init__(self, file):
        self._fd = file.fileno()
        self.buffer = BytesIO()
        self.writable = "x" in file.mode or "r" not in file.mode
        self.write = self.buffer.write if self.writable else None

    def read(self):
        while True:
            b = os.read(self._fd, max(os.fstat(self._fd).st_size, BUFSIZE))
            if not b:
                break
            ptr = self.buffer.tell()
            self.buffer.seek(0, 2), self.buffer.write(b), self.buffer.seek(ptr)
        self.newlines = 0
        return self.buffer.read()

    def readline(self):
        while self.newlines == 0:
            b = os.read(self._fd, max(os.fstat(self._fd).st_size, BUFSIZE))
            self.newlines = b.count(b"\n") + (not b)
            ptr = self.buffer.tell()
            self.buffer.seek(0, 2), self.buffer.write(b), self.buffer.seek(ptr)
        self.newlines -= 1
        return self.buffer.readline()

    def flush(self):
        if self.writable:
            os.write(self._fd, self.buffer.getvalue())
            self.buffer.truncate(0), self.buffer.seek(0)


class IOWrapper(IOBase):
    # pylint: disable=super-init-not-called
    def __init__(self, file):
        self.buffer = FastIO(file)
        self.flush = self.buffer.flush
        self.writable = self.buffer.writable
        self.write = lambda s: self.buffer.write(s.encode("ascii"))
        self.read = lambda: self.buffer.read().decode("ascii")
        self.readline = lambda: self.buffer.readline().decode("ascii")


def configure_log() -> None:
    "Configure the log output"
    log_formatter = logging.Formatter("L%(lineno)d - " "%(message)s")
    handler = logging.StreamHandler(IOWrapper(sys.stderr))
    handler.setFormatter(log_formatter)
    LOG.addHandler(handler)


LOG = None
# for interactive call: do not add multiple times the handler
if not LOG:
    LOG = logging.getLogger("template")
    configure_log()


def main(argv=None):
    "Program wrapper."
    if argv is None:
        argv = sys.argv[1:]
    parser = ArgumentParser()
    parser.add_argument(
        "-v",
        "--verbose",
        dest="verbose",
        action="store_true",
        default=False,
        help="run as verbose mode",
    )
    args = parser.parse_args(argv)
    if args.verbose:
        LOG.setLevel(logging.DEBUG)
    stdin = IOWrapper(sys.stdin)
    stdout = IOWrapper(sys.stdout)
    do_job(stdin, stdout)
    stdout.flush()
    for h in LOG.handlers:
        h.flush()
    return 0


if __name__ == "__main__":
    sys.exit(main())


Fixed submission:
#!/usr/bin/env python3
# vim: set fileencoding=utf-8

# pylint: disable=unused-import, invalid-name, missing-docstring, bad-continuation


"""Module docstring
"""

import functools
import heapq
import itertools
import logging
import math
import os
import random
import string
import sys
from argparse import ArgumentParser
from collections import defaultdict, deque
from copy import deepcopy
from io import BytesIO, IOBase
from typing import Dict, List, Optional, Set, Tuple


def solve(values: List[Tuple[int, int]], n: int) -> List[int]:
    passages = defaultdict(set)
    for a, b in values:
        passages[a].add(b)
        passages[b].add(a)
    if len(passages) < n:
        return []
    to_visit = [(0, 1, 0)]
    visited = set()
    # 0 idx is not used
    signs = [1] * (n + 1)
    while to_visit:
        min_dist, cur, prev = heapq.heappop(to_visit)
        if cur in visited:
            continue
        signs[cur] = prev
        for neighbour in passages[cur]:
            if neighbour in visited:
                continue
            heapq.heappush(to_visit, (min_dist + 1, neighbour, cur))
        visited.add(cur)
    LOG.debug((visited))
    LOG.debug((signs))
    if len(visited) < n:
        return []
    return signs[2:]


def do_job(stdin, stdout):
    "Do the work"
    LOG.debug("Start working")
    N, M = map(int, stdin.readline().split())
    if M < N - 1:
        print("No", file=stdout)
        return
    values = []
    for _ in range(M):
        a, b = map(int, stdin.readline().split())
        values.append((a, b))
    result = solve(values, N)
    if not result:
        print("No", file=stdout)
    else:
        print("Yes", file=stdout)
        print("\n".join(map(str, result)), file=stdout)


def print_output(testcase: int, result, stdout) -> None:
    "Formats and print result"
    if result is None:
        result = "IMPOSSIBLE"
    print("Case #{}: {}".format(testcase + 1, result), file=stdout)
    # 6 digits float precision {:.6f} (6 is the default value)
    # print("Case #{}: {:f}".format(testcase + 1, result), file=stdout)


BUFSIZE = 8192


class FastIO(IOBase):
    # pylint: disable=super-init-not-called, expression-not-assigned
    newlines = 0

    def __init__(self, file):
        self._fd = file.fileno()
        self.buffer = BytesIO()
        self.writable = "x" in file.mode or "r" not in file.mode
        self.write = self.buffer.write if self.writable else None

    def read(self):
        while True:
            b = os.read(self._fd, max(os.fstat(self._fd).st_size, BUFSIZE))
            if not b:
                break
            ptr = self.buffer.tell()
            self.buffer.seek(0, 2), self.buffer.write(b), self.buffer.seek(ptr)
        self.newlines = 0
        return self.buffer.read()

    def readline(self):
        while self.newlines == 0:
            b = os.read(self._fd, max(os.fstat(self._fd).st_size, BUFSIZE))
            self.newlines = b.count(b"\n") + (not b)
            ptr = self.buffer.tell()
            self.buffer.seek(0, 2), self.buffer.write(b), self.buffer.seek(ptr)
        self.newlines -= 1
        return self.buffer.readline()

    def flush(self):
        if self.writable:
            os.write(self._fd, self.buffer.getvalue())
            self.buffer.truncate(0), self.buffer.seek(0)


class IOWrapper(IOBase):
    # pylint: disable=super-init-not-called
    def __init__(self, file):
        self.buffer = FastIO(file)
        self.flush = self.buffer.flush
        self.writable = self.buffer.writable
        self.write = lambda s: self.buffer.write(s.encode("ascii"))
        self.read = lambda: self.buffer.read().decode("ascii")
        self.readline = lambda: self.buffer.readline().decode("ascii")


def configure_log() -> None:
    "Configure the log output"
    log_formatter = logging.Formatter("L%(lineno)d - " "%(message)s")
    handler = logging.StreamHandler(IOWrapper(sys.stderr))
    handler.setFormatter(log_formatter)
    LOG.addHandler(handler)


LOG = None
# for interactive call: do not add multiple times the handler
if not LOG:
    LOG = logging.getLogger("template")
    configure_log()


def main(argv=None):
    "Program wrapper."
    if argv is None:
        argv = sys.argv[1:]
    parser = ArgumentParser()
    parser.add_argument(
        "-v",
        "--verbose",
        dest="verbose",
        action="store_true",
        default=False,
        help="run as verbose mode",
    )
    args = parser.parse_args(argv)
    if args.verbose:
        LOG.setLevel(logging.DEBUG)
    stdin = IOWrapper(sys.stdin)
    stdout = IOWrapper(sys.stdout)
    do_job(stdin, stdout)
    stdout.flush()
    for h in LOG.handlers:
        h.flush()
    return 0


if __name__ == "__main__":
    sys.exit(main())

Buggy python submission (verdict: Time Limit Exceeded):
#!/usr/bin/env python3
# vim: set fileencoding=utf-8

# pylint: disable=unused-import, invalid-name, missing-docstring, bad-continuation


"""Module docstring
"""

import functools
import heapq
import itertools
import logging
import math
import os
import random
import string
import sys
from argparse import ArgumentParser
from collections import defaultdict, deque
from copy import deepcopy
from io import BytesIO, IOBase
from typing import Dict, List, Optional, Set, Tuple


def solve(values: List[Tuple[int, int]], n: int) -> List[int]:
    passages = defaultdict(set)
    for a, b in values:
        passages[a].add(b)
        passages[b].add(a)
    to_visit = deque((1,))
    visited = set()
    distances = defaultdict(lambda: sys.maxsize)
    distances[1] = 0
    signs = {}
    while to_visit:
        cur = to_visit.popleft()
        for neighbour in passages[cur]:
            tmp = distances[cur] + 1
            if distances[neighbour] > tmp:
                distances[neighbour] = distances[cur] + 1
                signs[neighbour] = cur
            if neighbour not in visited:
                to_visit.append(neighbour)
        visited.add(cur)
    # LOG.debug((visited, to_visit))
    if len(visited) < n:
        return []
    return [signs[i] for i in range(2, n + 1)]


def do_job(stdin, stdout):
    "Do the work"
    LOG.debug("Start working")
    N, M = map(int, stdin.readline().split())
    values = []
    for _ in range(M):
        a, b = map(int, stdin.readline().split())
        values.append((a, b))
    result = solve(values, N)
    if not result:
        print("No", file=stdout)
    else:
        print("Yes", file=stdout)
        print("\n".join(map(str, result)), file=stdout)


def print_output(testcase: int, result, stdout) -> None:
    "Formats and print result"
    if result is None:
        result = "IMPOSSIBLE"
    print("Case #{}: {}".format(testcase + 1, result), file=stdout)
    # 6 digits float precision {:.6f} (6 is the default value)
    # print("Case #{}: {:f}".format(testcase + 1, result), file=stdout)


BUFSIZE = 8192


class FastIO(IOBase):
    # pylint: disable=super-init-not-called, expression-not-assigned
    newlines = 0

    def __init__(self, file):
        self._fd = file.fileno()
        self.buffer = BytesIO()
        self.writable = "x" in file.mode or "r" not in file.mode
        self.write = self.buffer.write if self.writable else None

    def read(self):
        while True:
            b = os.read(self._fd, max(os.fstat(self._fd).st_size, BUFSIZE))
            if not b:
                break
            ptr = self.buffer.tell()
            self.buffer.seek(0, 2), self.buffer.write(b), self.buffer.seek(ptr)
        self.newlines = 0
        return self.buffer.read()

    def readline(self):
        while self.newlines == 0:
            b = os.read(self._fd, max(os.fstat(self._fd).st_size, BUFSIZE))
            self.newlines = b.count(b"\n") + (not b)
            ptr = self.buffer.tell()
            self.buffer.seek(0, 2), self.buffer.write(b), self.buffer.seek(ptr)
        self.newlines -= 1
        return self.buffer.readline()

    def flush(self):
        if self.writable:
            os.write(self._fd, self.buffer.getvalue())
            self.buffer.truncate(0), self.buffer.seek(0)


class IOWrapper(IOBase):
    # pylint: disable=super-init-not-called
    def __init__(self, file):
        self.buffer = FastIO(file)
        self.flush = self.buffer.flush
        self.writable = self.buffer.writable
        self.write = lambda s: self.buffer.write(s.encode("ascii"))
        self.read = lambda: self.buffer.read().decode("ascii")
        self.readline = lambda: self.buffer.readline().decode("ascii")


def configure_log() -> None:
    "Configure the log output"
    log_formatter = logging.Formatter("L%(lineno)d - " "%(message)s")
    handler = logging.StreamHandler(IOWrapper(sys.stderr))
    handler.setFormatter(log_formatter)
    LOG.addHandler(handler)


LOG = None
# for interactive call: do not add multiple times the handler
if not LOG:
    LOG = logging.getLogger("template")
    configure_log()


def main(argv=None):
    "Program wrapper."
    if argv is None:
        argv = sys.argv[1:]
    parser = ArgumentParser()
    parser.add_argument(
        "-v",
        "--verbose",
        dest="verbose",
        action="store_true",
        default=False,
        help="run as verbose mode",
    )
    args = parser.parse_args(argv)
    if args.verbose:
        LOG.setLevel(logging.DEBUG)
    stdin = IOWrapper(sys.stdin)
    stdout = IOWrapper(sys.stdout)
    do_job(stdin, stdout)
    stdout.flush()
    for h in LOG.handlers:
        h.flush()
    return 0


if __name__ == "__main__":
    sys.exit(main())


Fixed submission:
#!/usr/bin/env python3
# vim: set fileencoding=utf-8

# pylint: disable=unused-import, invalid-name, missing-docstring, bad-continuation


"""Module docstring
"""

import functools
import heapq
import itertools
import logging
import math
import os
import random
import string
import sys
from argparse import ArgumentParser
from collections import defaultdict, deque
from copy import deepcopy
from io import BytesIO, IOBase
from typing import Dict, List, Optional, Set, Tuple


def solve(values: List[Tuple[int, int]], n: int) -> List[int]:
    passages = defaultdict(set)
    for a, b in values:
        passages[a].add(b)
        passages[b].add(a)
    if len(passages) < n:
        return []
    to_visit = [(0, 1, 0)]
    visited = set()
    # 0 idx is not used
    signs = [1] * (n + 1)
    while to_visit:
        min_dist, cur, prev = heapq.heappop(to_visit)
        if cur in visited:
            continue
        signs[cur] = prev
        for neighbour in passages[cur]:
            if neighbour in visited:
                continue
            heapq.heappush(to_visit, (min_dist + 1, neighbour, cur))
        visited.add(cur)
    LOG.debug((visited))
    LOG.debug((signs))
    if len(visited) < n:
        return []
    return signs[2:]


def do_job(stdin, stdout):
    "Do the work"
    LOG.debug("Start working")
    N, M = map(int, stdin.readline().split())
    if M < N - 1:
        print("No", file=stdout)
        return
    values = []
    for _ in range(M):
        a, b = map(int, stdin.readline().split())
        values.append((a, b))
    result = solve(values, N)
    if not result:
        print("No", file=stdout)
    else:
        print("Yes", file=stdout)
        print("\n".join(map(str, result)), file=stdout)


def print_output(testcase: int, result, stdout) -> None:
    "Formats and print result"
    if result is None:
        result = "IMPOSSIBLE"
    print("Case #{}: {}".format(testcase + 1, result), file=stdout)
    # 6 digits float precision {:.6f} (6 is the default value)
    # print("Case #{}: {:f}".format(testcase + 1, result), file=stdout)


BUFSIZE = 8192


class FastIO(IOBase):
    # pylint: disable=super-init-not-called, expression-not-assigned
    newlines = 0

    def __init__(self, file):
        self._fd = file.fileno()
        self.buffer = BytesIO()
        self.writable = "x" in file.mode or "r" not in file.mode
        self.write = self.buffer.write if self.writable else None

    def read(self):
        while True:
            b = os.read(self._fd, max(os.fstat(self._fd).st_size, BUFSIZE))
            if not b:
                break
            ptr = self.buffer.tell()
            self.buffer.seek(0, 2), self.buffer.write(b), self.buffer.seek(ptr)
        self.newlines = 0
        return self.buffer.read()

    def readline(self):
        while self.newlines == 0:
            b = os.read(self._fd, max(os.fstat(self._fd).st_size, BUFSIZE))
            self.newlines = b.count(b"\n") + (not b)
            ptr = self.buffer.tell()
            self.buffer.seek(0, 2), self.buffer.write(b), self.buffer.seek(ptr)
        self.newlines -= 1
        return self.buffer.readline()

    def flush(self):
        if self.writable:
            os.write(self._fd, self.buffer.getvalue())
            self.buffer.truncate(0), self.buffer.seek(0)


class IOWrapper(IOBase):
    # pylint: disable=super-init-not-called
    def __init__(self, file):
        self.buffer = FastIO(file)
        self.flush = self.buffer.flush
        self.writable = self.buffer.writable
        self.write = lambda s: self.buffer.write(s.encode("ascii"))
        self.read = lambda: self.buffer.read().decode("ascii")
        self.readline = lambda: self.buffer.readline().decode("ascii")


def configure_log() -> None:
    "Configure the log output"
    log_formatter = logging.Formatter("L%(lineno)d - " "%(message)s")
    handler = logging.StreamHandler(IOWrapper(sys.stderr))
    handler.setFormatter(log_formatter)
    LOG.addHandler(handler)


LOG = None
# for interactive call: do not add multiple times the handler
if not LOG:
    LOG = logging.getLogger("template")
    configure_log()


def main(argv=None):
    "Program wrapper."
    if argv is None:
        argv = sys.argv[1:]
    parser = ArgumentParser()
    parser.add_argument(
        "-v",
        "--verbose",
        dest="verbose",
        action="store_true",
        default=False,
        help="run as verbose mode",
    )
    args = parser.parse_args(argv)
    if args.verbose:
        LOG.setLevel(logging.DEBUG)
    stdin = IOWrapper(sys.stdin)
    stdout = IOWrapper(sys.stdout)
    do_job(stdin, stdout)
    stdout.flush()
    for h in LOG.handlers:
        h.flush()
    return 0


if __name__ == "__main__":
    sys.exit(main())

Buggy python submission (verdict: Time Limit Exceeded):
#!/usr/bin/env python3
# vim: set fileencoding=utf-8

# pylint: disable=unused-import, invalid-name, missing-docstring, bad-continuation


"""Module docstring
"""

import functools
import heapq
import itertools
import logging
import math
import os
import random
import string
import sys
from argparse import ArgumentParser
from collections import defaultdict, deque
from copy import deepcopy
from io import BytesIO, IOBase
from typing import Dict, List, Optional, Set, Tuple


def solve(values: List[Tuple[int, int]], n: int) -> List[int]:
    passages = defaultdict(set)
    for a, b in values:
        passages[a].add(b)
        passages[b].add(a)
    to_visit = deque((1,))
    visited = set()
    distances = defaultdict(lambda: sys.maxsize)
    distances[1] = 0
    while to_visit:
        cur = to_visit.popleft()
        for neighbour in passages[cur]:
            distances[neighbour] = min(distances[neighbour], distances[cur] + 1)
            if neighbour not in visited:
                to_visit.append(neighbour)
        visited.add(cur)
    # LOG.debug((visited, to_visit))
    if len(visited) < n:
        return []
    result = []
    for i in range(2, n + 1):
        found = 1
        min_dist = sys.maxsize
        for neighbour in passages[i]:
            if distances[neighbour] < min_dist:
                found = neighbour
                min_dist = distances[neighbour]
        if min_dist == sys.maxsize:
            return []
        result.append(found)
    return result


def do_job(stdin, stdout):
    "Do the work"
    LOG.debug("Start working")
    N, M = map(int, stdin.readline().split())
    values = []
    for _ in range(M):
        a, b = map(int, stdin.readline().split())
        values.append((a, b))
    result = solve(values, N)
    if not result:
        print("No", file=stdout)
    else:
        print("Yes", file=stdout)
        print("\n".join(map(str, result)), file=stdout)


def print_output(testcase: int, result, stdout) -> None:
    "Formats and print result"
    if result is None:
        result = "IMPOSSIBLE"
    print("Case #{}: {}".format(testcase + 1, result), file=stdout)
    # 6 digits float precision {:.6f} (6 is the default value)
    # print("Case #{}: {:f}".format(testcase + 1, result), file=stdout)


BUFSIZE = 8192


class FastIO(IOBase):
    # pylint: disable=super-init-not-called, expression-not-assigned
    newlines = 0

    def __init__(self, file):
        self._fd = file.fileno()
        self.buffer = BytesIO()
        self.writable = "x" in file.mode or "r" not in file.mode
        self.write = self.buffer.write if self.writable else None

    def read(self):
        while True:
            b = os.read(self._fd, max(os.fstat(self._fd).st_size, BUFSIZE))
            if not b:
                break
            ptr = self.buffer.tell()
            self.buffer.seek(0, 2), self.buffer.write(b), self.buffer.seek(ptr)
        self.newlines = 0
        return self.buffer.read()

    def readline(self):
        while self.newlines == 0:
            b = os.read(self._fd, max(os.fstat(self._fd).st_size, BUFSIZE))
            self.newlines = b.count(b"\n") + (not b)
            ptr = self.buffer.tell()
            self.buffer.seek(0, 2), self.buffer.write(b), self.buffer.seek(ptr)
        self.newlines -= 1
        return self.buffer.readline()

    def flush(self):
        if self.writable:
            os.write(self._fd, self.buffer.getvalue())
            self.buffer.truncate(0), self.buffer.seek(0)


class IOWrapper(IOBase):
    # pylint: disable=super-init-not-called
    def __init__(self, file):
        self.buffer = FastIO(file)
        self.flush = self.buffer.flush
        self.writable = self.buffer.writable
        self.write = lambda s: self.buffer.write(s.encode("ascii"))
        self.read = lambda: self.buffer.read().decode("ascii")
        self.readline = lambda: self.buffer.readline().decode("ascii")


def configure_log() -> None:
    "Configure the log output"
    log_formatter = logging.Formatter("L%(lineno)d - " "%(message)s")
    handler = logging.StreamHandler(IOWrapper(sys.stderr))
    handler.setFormatter(log_formatter)
    LOG.addHandler(handler)


LOG = None
# for interactive call: do not add multiple times the handler
if not LOG:
    LOG = logging.getLogger("template")
    configure_log()


def main(argv=None):
    "Program wrapper."
    if argv is None:
        argv = sys.argv[1:]
    parser = ArgumentParser()
    parser.add_argument(
        "-v",
        "--verbose",
        dest="verbose",
        action="store_true",
        default=False,
        help="run as verbose mode",
    )
    args = parser.parse_args(argv)
    if args.verbose:
        LOG.setLevel(logging.DEBUG)
    stdin = IOWrapper(sys.stdin)
    stdout = IOWrapper(sys.stdout)
    do_job(stdin, stdout)
    stdout.flush()
    for h in LOG.handlers:
        h.flush()
    return 0


if __name__ == "__main__":
    sys.exit(main())


Fixed submission:
#!/usr/bin/env python3
# vim: set fileencoding=utf-8

# pylint: disable=unused-import, invalid-name, missing-docstring, bad-continuation


"""Module docstring
"""

import functools
import heapq
import itertools
import logging
import math
import os
import random
import string
import sys
from argparse import ArgumentParser
from collections import defaultdict, deque
from copy import deepcopy
from io import BytesIO, IOBase
from typing import Dict, List, Optional, Set, Tuple


def solve(values: List[Tuple[int, int]], n: int) -> List[int]:
    passages = defaultdict(set)
    for a, b in values:
        passages[a].add(b)
        passages[b].add(a)
    if len(passages) < n:
        return []
    to_visit = [(0, 1, 0)]
    visited = set()
    # 0 idx is not used
    signs = [1] * (n + 1)
    while to_visit:
        min_dist, cur, prev = heapq.heappop(to_visit)
        if cur in visited:
            continue
        signs[cur] = prev
        for neighbour in passages[cur]:
            if neighbour in visited:
                continue
            heapq.heappush(to_visit, (min_dist + 1, neighbour, cur))
        visited.add(cur)
    LOG.debug((visited))
    LOG.debug((signs))
    if len(visited) < n:
        return []
    return signs[2:]


def do_job(stdin, stdout):
    "Do the work"
    LOG.debug("Start working")
    N, M = map(int, stdin.readline().split())
    if M < N - 1:
        print("No", file=stdout)
        return
    values = []
    for _ in range(M):
        a, b = map(int, stdin.readline().split())
        values.append((a, b))
    result = solve(values, N)
    if not result:
        print("No", file=stdout)
    else:
        print("Yes", file=stdout)
        print("\n".join(map(str, result)), file=stdout)


def print_output(testcase: int, result, stdout) -> None:
    "Formats and print result"
    if result is None:
        result = "IMPOSSIBLE"
    print("Case #{}: {}".format(testcase + 1, result), file=stdout)
    # 6 digits float precision {:.6f} (6 is the default value)
    # print("Case #{}: {:f}".format(testcase + 1, result), file=stdout)


BUFSIZE = 8192


class FastIO(IOBase):
    # pylint: disable=super-init-not-called, expression-not-assigned
    newlines = 0

    def __init__(self, file):
        self._fd = file.fileno()
        self.buffer = BytesIO()
        self.writable = "x" in file.mode or "r" not in file.mode
        self.write = self.buffer.write if self.writable else None

    def read(self):
        while True:
            b = os.read(self._fd, max(os.fstat(self._fd).st_size, BUFSIZE))
            if not b:
                break
            ptr = self.buffer.tell()
            self.buffer.seek(0, 2), self.buffer.write(b), self.buffer.seek(ptr)
        self.newlines = 0
        return self.buffer.read()

    def readline(self):
        while self.newlines == 0:
            b = os.read(self._fd, max(os.fstat(self._fd).st_size, BUFSIZE))
            self.newlines = b.count(b"\n") + (not b)
            ptr = self.buffer.tell()
            self.buffer.seek(0, 2), self.buffer.write(b), self.buffer.seek(ptr)
        self.newlines -= 1
        return self.buffer.readline()

    def flush(self):
        if self.writable:
            os.write(self._fd, self.buffer.getvalue())
            self.buffer.truncate(0), self.buffer.seek(0)


class IOWrapper(IOBase):
    # pylint: disable=super-init-not-called
    def __init__(self, file):
        self.buffer = FastIO(file)
        self.flush = self.buffer.flush
        self.writable = self.buffer.writable
        self.write = lambda s: self.buffer.write(s.encode("ascii"))
        self.read = lambda: self.buffer.read().decode("ascii")
        self.readline = lambda: self.buffer.readline().decode("ascii")


def configure_log() -> None:
    "Configure the log output"
    log_formatter = logging.Formatter("L%(lineno)d - " "%(message)s")
    handler = logging.StreamHandler(IOWrapper(sys.stderr))
    handler.setFormatter(log_formatter)
    LOG.addHandler(handler)


LOG = None
# for interactive call: do not add multiple times the handler
if not LOG:
    LOG = logging.getLogger("template")
    configure_log()


def main(argv=None):
    "Program wrapper."
    if argv is None:
        argv = sys.argv[1:]
    parser = ArgumentParser()
    parser.add_argument(
        "-v",
        "--verbose",
        dest="verbose",
        action="store_true",
        default=False,
        help="run as verbose mode",
    )
    args = parser.parse_args(argv)
    if args.verbose:
        LOG.setLevel(logging.DEBUG)
    stdin = IOWrapper(sys.stdin)
    stdout = IOWrapper(sys.stdout)
    do_job(stdin, stdout)
    stdout.flush()
    for h in LOG.handlers:
        h.flush()
    return 0


if __name__ == "__main__":
    sys.exit(main())