

# A Bipartite Graph is a graph whose vertices can be divided into two independent sets,
# U and V such that every edge (u, v) either connects a vertex from U to V or a vertex
# from V to U. In other words, for every edge (u, v), either u belongs to U and v to V,
# or u belongs to V and v to U. We can also say that there is no edge that connects
# vertices of same set.
from queue import Queue



def    UpperCamelCase(					lowercase_							)	->				Union[str, Any]:





  '''simple docstring'''





  snake_case_       						=      Queue()
  snake_case_       						=      [False] * len(lowercase_							)
  snake_case_       						=      [-1] * len(lowercase_							)

  def bfs():
    while not queue.empty():
      snake_case_       						=      queue.get()
      snake_case_       						=      True

      for neighbour in graph[u]:
        if neighbour == u:
          return False

        if color[neighbour] == -1:
          snake_case_       						=      1 - color[u]
          queue.put(lowercase_							)

        elif color[neighbour] == color[u]:
          return False

    return True

  for i in range(len(lowercase_							)							):
    if not visited[i]:
      queue.put(lowercase_							)
      snake_case_       						=      0
      if bfs() is False:
        return False

  return True


if __name__ == "__main__":
    # Adjacency List of graph
    print(check_bipartite({0: [1, 3], 1: [0, 2], 2: [1, 3], 3: [0, 2]}))


def    UpperCamelCase(					lowercase_							)	->				str:





  '''simple docstring'''





  if len(lowercase_							) == 0 or not all(len(lowercase_							) == 2 for pair in coordinates							):
    raise ValueError("""The program cannot work out a fitting polynomial."""							)

  if len({tuple(lowercase_							) for pair in coordinates}							) != len(lowercase_							):
    raise ValueError("""The program cannot work out a fitting polynomial."""							)

  snake_case_       						=      {x for x, _ in coordinates}
  if len(lowercase_							) == 1:
    return f'''x={coordinates[0][0]}'''

  if len(lowercase_							) != len(lowercase_							):
    raise ValueError("""The program cannot work out a fitting polynomial."""							)

  snake_case_       						=      len(lowercase_							)

  snake_case_       						=      0
  snake_case_       						=      []
  # put the x and x to the power values in a matrix
  while count_of_line < x:
    snake_case_       						=      0
    snake_case_       						=      coordinates[count_of_line][0]
    snake_case_       						=      []
    while count_in_line < x:
      count_line.append(a ** (x - (count_in_line + 1))							)
      count_in_line += 1
    matrix.append(lowercase_							)
    count_of_line += 1

  snake_case_       						=      0
  # put the y values into a vector
  snake_case_       						=      []
  while count_of_line < x:
    vector.append(coordinates[count_of_line][1]							)
    count_of_line += 1

  snake_case_       						=      0

  while count < x:
    snake_case_       						=      0
    while zahlen < x:
      if count == zahlen:
        zahlen += 1
      if zahlen == x:
        break
      snake_case_       						=      matrix[zahlen][count] / matrix[count][count]
      for counting_columns, item in enumerate(matrix[count]							):
        # manipulating all the values in the matrix
        matrix[zahlen][counting_columns] -= item * bruch
      # manipulating the values in the vector
      vector[zahlen] -= vector[count] * bruch
      zahlen += 1
    count += 1

  snake_case_       						=      0
  # make solutions
  snake_case_       						=      []
  while count < x:
    solution.append(str(vector[count] / matrix[count][count]							)							)
    count += 1

  snake_case_       						=      0
  snake_case_       						=      """f(x)="""

  while count < x:
    snake_case_       						=      solution[count].split("""E"""							)
    if len(lowercase_							) > 1:
      snake_case_       						=      f'''{remove_e[0]}*10^{remove_e[1]}'''
    solved += f'''x^{x - (count + 1)}*{solution[count]}'''
    if count + 1 != x:
      solved += "+"
    count += 1

  return solved


if __name__ == "__main__":
    print(points_to_polynomial([]))
    print(points_to_polynomial([[]]))
    print(points_to_polynomial([[1, 0], [2, 0], [3, 0]]))
    print(points_to_polynomial([[1, 1], [2, 1], [3, 1]]))
    print(points_to_polynomial([[1, 3], [2, 3], [3, 3]]))
    print(points_to_polynomial([[1, 1], [2, 2], [3, 3]]))
    print(points_to_polynomial([[1, 1], [2, 4], [3, 9]]))
    print(points_to_polynomial([[1, 3], [2, 6], [3, 11]]))
    print(points_to_polynomial([[1, -3], [2, -6], [3, -11]]))
    print(points_to_polynomial([[1, 5], [2, 2], [3, 9]]))