code: 
  
  
  
  
  
  
  """simple docstring"""
  
  from __future__ import annotations
  
  from collections import deque
  from enum import Enum
  from math import atana, degrees
  from sys import maxsize
  
  
  class 	__lowerCamelCase       (				A__    ):
  
  
  
  
  
  		'''simple docstring'''
  
  
  
  
  
  
  		a_			:       List[str]      =							1
  		a_			:       Optional[Any]      =							2
  		a_			:       Dict      =							3
  
  
  
  
  
  		def __repr__(     self							:			Any			):
  				return f'''{self.__class__.__name__}.{self.name}'''
  
  
  
  
  
  
  
  def 	__lowerCamelCase			(						__UpperCamelCase   , __UpperCamelCase   , __UpperCamelCase )		->					float:
  
  		"""simple docstring"""
  
  
  
  
  
  
  
  		lowerCAmelCase_       ,			lowerCAmelCase_     :							Union[str, Any]             =    point
  		return degrees(atana(y - miny   , x - minx ) )
  
  def 	__lowerCamelCase			(						__UpperCamelCase   , __UpperCamelCase   , __UpperCamelCase )		->					Direction:
  
  		"""simple docstring"""
  
  
  
  
  
  
  
  		lowerCAmelCase_       ,			lowerCAmelCase_     :							Tuple             =    starting
  		lowerCAmelCase_       ,			lowerCAmelCase_     :							Any             =    via
  		lowerCAmelCase_       ,			lowerCAmelCase_     :							str             =    target
  		lowerCAmelCase_     :							List[Any]             =    degrees(atana(ya - ya   , xa - xa ) )
  		via_angle %= 360
  		lowerCAmelCase_     :							Optional[Any]             =    degrees(atana(ya - ya   , xa - xa ) )
  		target_angle %= 360
  		# t-
  		#  \ \
  		#   \ v
  		#    \|
  		#     s
  		# via_angle is always lower than target_angle, if direction is left.
  		# If they are same, it means they are on a same line of convex hull.
  		if target_angle > via_angle:
  				return Direction.left
  		elif target_angle == via_angle:
  				return Direction.straight
  		else:
  				return Direction.right
  
  
  
  def 	__lowerCamelCase			(						__UpperCamelCase )		->					list[tuple[int, int]]:
  
  		"""simple docstring"""
  
  
  
  
  
  
  
  		if len(__UpperCamelCase ) <= 2:
  				# There is no convex hull
  				raise ValueError("graham_scan: argument must contain more than 3 points." )
  		if len(__UpperCamelCase ) == 3:
  				return points
  		# find the lowest and the most left point
  		lowerCAmelCase_     :							Optional[int]             =    0
  		lowerCAmelCase_       ,			lowerCAmelCase_     :							List[Any]             =    maxsize, maxsize
  		for i, point in enumerate(__UpperCamelCase ):
  				lowerCAmelCase_     :							List[str]             =    point[0]
  				lowerCAmelCase_     :							Optional[Any]             =    point[1]
  				if y < miny:
  						lowerCAmelCase_     :							Union[str, Any]             =    y
  						lowerCAmelCase_     :							Union[str, Any]             =    x
  						lowerCAmelCase_     :							List[Any]             =    i
  				if y == miny and x < minx:
  						lowerCAmelCase_     :							str             =    x
  						lowerCAmelCase_     :							Union[str, Any]             =    i
  
      # remove the lowest and the most left point from points for preparing for sort
  		points.pop(__UpperCamelCase )
  
  		lowerCAmelCase_     :							Optional[int]             =    sorted(__UpperCamelCase   , key=lambda __UpperCamelCase : angle_comparer(__UpperCamelCase   , __UpperCamelCase   , __UpperCamelCase ) )
  		# This insert actually costs complexity,
  		# and you should instead add (minx, miny) into stack later.
  		# I'm using insert just for easy understanding.
  		sorted_points.insert(0   , (minx, miny) )
  
  		lowerCAmelCase_     :							deque[tuple[int, int]]             =    deque()
  		stack.append(sorted_points[0] )
  		stack.append(sorted_points[1] )
  		stack.append(sorted_points[2] )
  		# In any ways, the first 3 points line are towards left.
  		# Because we sort them the angle from minx, miny.
  		lowerCAmelCase_     :							Dict             =    Direction.left
  
  		for i in range(3   , len(__UpperCamelCase ) ):
  				while True:
  						lowerCAmelCase_     :							Any             =    stack[-2]
  						lowerCAmelCase_     :							int             =    stack[-1]
  						lowerCAmelCase_     :							Dict             =    sorted_points[i]
  						lowerCAmelCase_     :							List[Any]             =    check_direction(__UpperCamelCase   , __UpperCamelCase   , __UpperCamelCase )
  
  						if next_direction == Direction.left:
  								lowerCAmelCase_     :							List[str]             =    Direction.left
  								break
  						if next_direction == Direction.straight:
  								if current_direction == Direction.left:
  										# We keep current_direction as left.
  										# Because if the straight line keeps as straight,
  										# we want to know if this straight line is towards left.
  										break
  								elif current_direction == Direction.right:
  										# If the straight line is towards right,
  										# every previous points on those straigh line is not convex hull.
  										stack.pop()
  						if next_direction == Direction.right:
  								stack.pop()
  				stack.append(sorted_points[i] )
  		return list(__UpperCamelCase )
  
style_context: 
  
  
  
  
  
  
  """simple docstring"""
  
  from __future__ import annotations
  
  import random
  
  # Adjacency list representation of this graph:
  # https://en.wikipedia.org/wiki/File:Single_run_of_Karger%E2%80%99s_Mincut_algorithm.svg
  lowercase__      =				{
      """1""": ["""2""", """3""", """4""", """5"""],
      """2""": ["""1""", """3""", """4""", """5"""],
      """3""": ["""1""", """2""", """4""", """5""", """10"""],
      """4""": ["""1""", """2""", """3""", """5""", """6"""],
      """5""": ["""1""", """2""", """3""", """4""", """7"""],
      """6""": ["""7""", """8""", """9""", """10""", """4"""],
      """7""": ["""6""", """8""", """9""", """10""", """5"""],
      """8""": ["""6""", """7""", """9""", """10"""],
      """9""": ["""6""", """7""", """8""", """10"""],
      """10""": ["""6""", """7""", """8""", """9""", """3"""],
  }
  
  def 	__lowerCamelCase			(						__UpperCamelCase )		->					set[tuple[str, str]]:
  
  		"""simple docstring"""
  
  
  
  
  
  
  
  		lowerCAmelCase_     :							Union[str, Any]             =    {node: {node} for node in graph}
  
  		lowerCAmelCase_     :							List[str]             =    {node: graph[node][:] for node in graph}
  
  		while len(__UpperCamelCase ) > 2:
  				# Choose a random edge.
  				lowerCAmelCase_     :							int             =    random.choice(list(graph_copy.keys() ) )
  				lowerCAmelCase_     :							Any             =    random.choice(graph_copy[u] )
  
  				# Contract edge (u, v) to new node uv
  				lowerCAmelCase_     :							Union[str, Any]             =    u + v
  				lowerCAmelCase_     :							List[Any]             =    list(set(graph_copy[u] + graph_copy[v] ) )
  				uv_neighbors.remove(__UpperCamelCase )
  				uv_neighbors.remove(__UpperCamelCase )
  				lowerCAmelCase_     :							str             =    uv_neighbors
  				for neighbor in uv_neighbors:
  						graph_copy[neighbor].append(__UpperCamelCase )
  
  				lowerCAmelCase_     :							Optional[int]             =    set(contracted_nodes[u].union(contracted_nodes[v] ) )
  
  				# Remove nodes u and v.
  				del graph_copy[u]
  				del graph_copy[v]
  				for neighbor in uv_neighbors:
  						if u in graph_copy[neighbor]:
  								graph_copy[neighbor].remove(__UpperCamelCase )
  						if v in graph_copy[neighbor]:
  								graph_copy[neighbor].remove(__UpperCamelCase )
  
      # Find cutset.
  		lowerCAmelCase_     :							Union[str, Any]             =    [contracted_nodes[node] for node in graph_copy]
  		return {
  		    (node, neighbor)
  		    for node in groups[0]
  		    for neighbor in graph[node]
  		    if neighbor in groups[1]
  		}
  
  
  if __name__ == "__main__":
  					print(partition_graph(TEST_GRAPH))
  