<commit_before>"""
    A Hamiltonian cycle (Hamiltonian circuit) is a graph cycle
    through a graph that visits each node exactly once.
    Determining whether such paths and cycles exist in graphs
    is the 'Hamiltonian path problem', which is NP-complete.

    Wikipedia: https://en.wikipedia.org/wiki/Hamiltonian_path
"""


def valid_connection(
    graph: list[list[int]], next_ver: int, curr_ind: int, path: list[int]
) -> bool:
    """
    Checks whether it is possible to add next into path by validating 2 statements
    1. There should be path between current and next vertex
    2. Next vertex should not be in path
    If both validations succeeds we return True saying that it is possible to connect
    this vertices either we return False

    Case 1:Use exact graph as in main function, with initialized values
    >>> graph = [[0, 1, 0, 1, 0],
    ...          [1, 0, 1, 1, 1],
    ...          [0, 1, 0, 0, 1],
    ...          [1, 1, 0, 0, 1],
    ...          [0, 1, 1, 1, 0]]
    >>> path = [0, -1, -1, -1, -1, 0]
    >>> curr_ind = 1
    >>> next_ver = 1<commit_msg>Correct grammar of comment in backtracking/hamiltonian_cycle.py (#4868)<commit_after>"""
    A Hamiltonian cycle (Hamiltonian circuit) is a graph cycle
    through a graph that visits each node exactly once.
    Determining whether such paths and cycles exist in graphs
    is the 'Hamiltonian path problem', which is NP-complete.

    Wikipedia: https://en.wikipedia.org/wiki/Hamiltonian_path
"""


def valid_connection(
    graph: list[list[int]], next_ver: int, curr_ind: int, path: list[int]
) -> bool:
    """
    Checks whether it is possible to add next into path by validating 2 statements
    1. There should be path between current and next vertex
    2. Next vertex should not be in path
    If both validations succeed we return True, saying that it is possible to connect
    this vertices, otherwise we return False

    Case 1:Use exact graph as in main function, with initialized values
    >>> graph = [[0, 1, 0, 1, 0],
    ...          [1, 0, 1, 1, 1],
    ...          [0, 1, 0, 0, 1],
    ...          [1, 1, 0, 0, 1],
    ...          [0, 1, 1, 1, 0]]
    >>> path = [0, -1, -1, -1, -1, 0]
    >>> curr_ind = 1
    >>> next_ver = 1
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