





def   lowerCamelCase__       (       UpperCamelCase__     :    list[list[int]]    ,      UpperCamelCase__     :    int    ,      UpperCamelCase__     :    int    ,      UpperCamelCase__     :    list[int]  )     ->	bool:



					'''simple docstring'''




					if graph[path[curr_ind - 1]][next_ver] == 0:
										return False

					# 2. Validate that next vertex is not already in path
					return not any(vertex == next_ver for vertex in path  )





def   lowerCamelCase__       (       UpperCamelCase__     :    list[list[int]]    ,      UpperCamelCase__     :    list[int]    ,      UpperCamelCase__     :    int  )     ->	bool:



					'''simple docstring'''




					if curr_ind == len(UpperCamelCase__  ):
										# return whether path exists between current and starting vertices
										return graph[path[curr_ind - 1]][path[0]] == 1

					# Recursive Step
					for next_ver in range(0    ,      len(UpperCamelCase__  )  ):
										if valid_connection(UpperCamelCase__    ,      UpperCamelCase__    ,      UpperCamelCase__    ,      UpperCamelCase__  ):
															# Insert current vertex  into path as next transition
															_snake_case													=			next_ver
															# Validate created path
															if util_hamilton_cycle(UpperCamelCase__    ,      UpperCamelCase__    ,      curr_ind + 1  ):
																				return True
															# Backtrack
															_snake_case													=			-1
					return False





def   lowerCamelCase__       (       UpperCamelCase__     :    list[list[int]]    ,      UpperCamelCase__     :    int = 0  )     ->	list[int]:



					'''simple docstring'''




					_snake_case													=			[-1] * (len(UpperCamelCase__  ) + 1)
					# initialize start and end of path with starting index
					_snake_case													=			_snake_case													=			start_index
					# evaluate and if we find answer return path either return empty array
					return path if util_hamilton_cycle(UpperCamelCase__    ,      UpperCamelCase__    ,      1  ) else []










def   lowerCamelCase__       (       UpperCamelCase__     :    list[list[int]]    ,      UpperCamelCase__     :    int    ,      UpperCamelCase__     :    int    ,      UpperCamelCase__     :    list[int]  )     ->	bool:



					'''simple docstring'''




					if graph[path[curr_ind - 1]][next_ver] == 0:
										return False

					# 2. Validate that next vertex is not already in path
					return not any(vertex == next_ver for vertex in path  )





def   lowerCamelCase__       (       UpperCamelCase__     :    list[list[int]]    ,      UpperCamelCase__     :    list[int]    ,      UpperCamelCase__     :    int  )     ->	bool:



					'''simple docstring'''




					if curr_ind == len(UpperCamelCase__  ):
										# return whether path exists between current and starting vertices
										return graph[path[curr_ind - 1]][path[0]] == 1

					# Recursive Step
					for next_ver in range(0    ,      len(UpperCamelCase__  )  ):
										if valid_connection(UpperCamelCase__    ,      UpperCamelCase__    ,      UpperCamelCase__    ,      UpperCamelCase__  ):
															# Insert current vertex  into path as next transition
															_snake_case													=			next_ver
															# Validate created path
															if util_hamilton_cycle(UpperCamelCase__    ,      UpperCamelCase__    ,      curr_ind + 1  ):
																				return True
															# Backtrack
															_snake_case													=			-1
					return False





def   lowerCamelCase__       (       UpperCamelCase__     :    list[list[int]]    ,      UpperCamelCase__     :    int = 0  )     ->	list[int]:



					'''simple docstring'''




					_snake_case													=			[-1] * (len(UpperCamelCase__  ) + 1)
					# initialize start and end of path with starting index
					_snake_case													=			_snake_case													=			start_index
					# evaluate and if we find answer return path either return empty array
					return path if util_hamilton_cycle(UpperCamelCase__    ,      UpperCamelCase__    ,      1  ) else []



