




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
__SCREAMING_SNAKE_CASE			:      Any      			={
    """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_	(     lowerCamelCase:				dict[str, list[str]]  )    ->	set[tuple[str, str]]:


      '''simple docstring'''




      lowerCamelCase__					     = {node: {node} for node in graph}

      lowerCamelCase__					     = {node: graph[node][:] for node in graph}

      while len(lowerCamelCase  ) > 2:
            # Choose a random edge.
            lowerCamelCase__					     = random.choice(list(graph_copy.keys()  )  )
            lowerCamelCase__					     = random.choice(graph_copy[u]  )

            # Contract edge (u, v) to new node uv
            lowerCamelCase__					     = u + v
            lowerCamelCase__					     = list(set(graph_copy[u] + graph_copy[v]  )  )
            uv_neighbors.remove(lowerCamelCase  )
            uv_neighbors.remove(lowerCamelCase  )
            lowerCamelCase__					     = uv_neighbors
            for neighbor in uv_neighbors:
                  graph_copy[neighbor].append(lowerCamelCase  )

            lowerCamelCase__					     = 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(lowerCamelCase  )
                  if v in graph_copy[neighbor]:
                        graph_copy[neighbor].remove(lowerCamelCase  )

    # Find cutset.
      lowerCamelCase__					     = [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))





def 	lowerCAmelCase_	(     lowerCamelCase:				int  )    ->	int:


      '''simple docstring'''




      return n * (2 * n - 1)

def 	lowerCAmelCase_	(     lowerCamelCase:				int  )    ->	bool:


      '''simple docstring'''




      lowerCamelCase__					     = (1 + 24 * n) ** 0.5
      return ((1 + root) / 6) % 1 == 0

def 	lowerCAmelCase_	(     lowerCamelCase:				int = 1_44  )    ->	int:


      '''simple docstring'''




      lowerCamelCase__					     = start
      lowerCamelCase__					     = hexagonal_num(lowerCamelCase  )
      while not is_pentagonal(lowerCamelCase  ):
            n += 1
            lowerCamelCase__					     = hexagonal_num(lowerCamelCase  )
      return num


if __name__ == "__main__":
      print(F"""{solution()} = """)