






"""simple docstring"""

from __future__ import annotations

from queue import Queue


class 	__lowerCamelCase       :





		'''simple docstring'''






		def __init__(     self							:			int			):
				lowerCAmelCase_     :							dict[int, list[int]]             =    {}





		def      lowerCamelCase (     self							:			List[str]			):
				for i in self.vertices:
						print(a_		,    " : "		,    " -> ".join([str(a_			) for j in self.vertices[i]]			)			)





		def      lowerCamelCase (     self							:			Optional[int]		,    a_							:			int		,    a_							:			int			):
				if from_vertex in self.vertices:
						self.vertices[from_vertex].append(a_			)
				else:
						lowerCAmelCase_     :							Any             =    [to_vertex]





		def      lowerCamelCase (     self							:			Union[str, Any]		,    a_							:			int			):
				lowerCAmelCase_     :							Optional[Any]             =    set()

				# create a first in first out queue to store all the vertices for BFS
				lowerCAmelCase_     :							Queue             =    Queue()

				# mark the source node as visited and enqueue it
				visited.add(a_			)
				queue.put(a_			)

				while not queue.empty():
						lowerCAmelCase_     :							List[str]             =    queue.get()

						# loop through all adjacent vertex and enqueue it if not yet visited
						for adjacent_vertex in self.vertices[vertex]:
								if adjacent_vertex not in visited:
										queue.put(a_			)
										visited.add(a_			)
				return visited


if __name__ == "__main__":
					from doctest import testmod

					testmod(verbose=True)

					lowercase__      =				Graph()
					g.add_edge(0, 1)
					g.add_edge(0, 2)
					g.add_edge(1, 2)
					g.add_edge(2, 0)
					g.add_edge(2, 3)
					g.add_edge(3, 3)

					g.print_graph()
					# 0  :  1 -> 2
					# 1  :  2
					# 2  :  0 -> 3
					# 3  :  3

					assert sorted(g.bfs(2)) == [0, 1, 2, 3]








"""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))
