
"""simple docstring"""
# using dfs for finding eulerian path traversal






def 				a__   (   a   :				Any       ,     a   :				Union[str, Any]       ,     a   :				Union[str, Any]       ,     a   :				Tuple=None		):

		"""simple docstring"""







		_snake_case     :						List[str]						=   (path or []) + [u]
		for v in graph[u]:
				if visited_edge[u][v] is False:
						_snake_case			,					_snake_case     :						Optional[Any]						=   True, True
						_snake_case     :						Optional[Any]						=   dfs(snake_case__       ,     snake_case__       ,     snake_case__       ,     snake_case__		)
		return path
def 				a__   (   a   :				List[Any]       ,     a   :				str		):

		"""simple docstring"""







		_snake_case     :						str						=   0
		_snake_case     :						Union[str, Any]						=   -1
		for i in range(snake_case__		):
				if i not in graph.keys():
						continue
				if len(graph[i]		) % 2 == 1:
						odd_degree_nodes += 1
						_snake_case     :						str						=   i
		if odd_degree_nodes == 0:
				return 1, odd_node
		if odd_degree_nodes == 2:
				return 2, odd_node
		return 3, odd_node
def 				a__   (   a   :				Union[str, Any]       ,     a   :				Any		):

		"""simple docstring"""







		_snake_case     :						Optional[int]						=   [[False for _ in range(max_node + 1		)] for _ in range(max_node + 1		)]
		_snake_case			,					_snake_case     :						Dict						=   check_circuit_or_path(snake_case__       ,     snake_case__		)
		if check == 3:
				print("graph is not Eulerian"		)
				print("no path"		)
				return
		_snake_case     :						Optional[Any]						=   1
		if check == 2:
				_snake_case     :						int						=   odd_node
				print("graph has a Euler path"		)
		if check == 1:
				print("graph has a Euler cycle"		)
		_snake_case     :						Optional[Any]						=   dfs(snake_case__       ,     snake_case__       ,     snake_case__		)
		print(snake_case__		)
def 				a__   (   ):

		"""simple docstring"""







		_snake_case     :						List[str]						=   {1: [2, 3, 4], 2: [1, 3], 3: [1, 2], 4: [1, 5], 5: [4]}
		_snake_case     :						int						=   {1: [2, 3, 4, 5], 2: [1, 3], 3: [1, 2], 4: [1, 5], 5: [1, 4]}
		_snake_case     :						Optional[Any]						=   {1: [2, 3, 4], 2: [1, 3, 4], 3: [1, 2], 4: [1, 2, 5], 5: [4]}
		_snake_case     :						Dict						=   {1: [2, 3], 2: [1, 3], 3: [1, 2]}
		_snake_case     :						Optional[int]						=   {
		    1: [],
		    2: []
		    # all degree is zero
		}
		_snake_case     :						Tuple						=   10
		check_euler(snake_case__       ,     snake_case__		)
		check_euler(snake_case__       ,     snake_case__		)
		check_euler(snake_case__       ,     snake_case__		)
		check_euler(snake_case__       ,     snake_case__		)
		check_euler(snake_case__       ,     snake_case__		)


if __name__ == "__main__":
				main()



"""simple docstring"""
def 				a__   (   a   :				list       ,     a   :				int       ,     a   :				int = 0       ,     a   :				int = 0		):

  """simple docstring"""







  _snake_case     :						Optional[int]						=   right or len(a		) - 1
  if left > right:
    return -1
  elif list_data[left] == key:
    return left
  elif list_data[right] == key:
    return right
  else:
    return search(a       ,     a       ,     left + 1       ,     right - 1		)


if __name__ == "__main__":
    import doctest

    doctest.testmod()

