from collections import deque
import random as rand
import math as math

# the dfault weight is 1 if not assigend but all the implementation is weighted

class DirectedGraph:
	def __init__(self):
		self.graph = {}

	# adding vertices and edges
	# adding the weight is optional
	# handels repetition
	def add_pair(self, u, v, w = 1):
		if self.graph.get(u):
			if self.graph[u].count([w,v]) == 0:
				self.graph[u].append([w, v])
		else:
			self.graph[u] = [[w, v]]
		if not self.graph.get(v):
			self.graph[v] = []
			
	# handels if the input does not exist
	def remove_pair(self, u, v):
		if self.graph.get(u):
			for _ in self.graph[u]:
				if _[1] == v:
					self.graph[u].remove(_)

	# if no destination is meant the defaut value is -1
	def dfs(self, s = -2, d = -1):
		if s == d:
			return []
		stack = []
		visited = []
		if s == -2:
			s = list(self.graph.keys())[0]
		stack.append(s)
		visited.append(s)
		ss = s

		while True:
			# check if there is any non isolated nodes
			if len(self.graph[s]) != 0:
				ss = s
				for __ in self.graph[s]:
					if visited.count(__[1]) < 1:
						if __[1] == d:
							visited.append(d)
							return visited
						else:
							stack.append(__[1])
							visited.append(__[1])
							ss =__[1]
							break

			# check if all the children are visited
			if s == ss :
				stack.pop()
				if len(stack) != 0:
					s = stack[len(stack) - 1]
			else:
				s = ss

			# check if se have reached the starting point
			if len(stack) == 0:
				return visited	

	# c is the count of nodes you want and if you leave it or pass -1 to the funtion the count
	# will be random from 10 to 10000
	def fill_graph_randomly(self, c = -1):
		if c == -1:
			c = (math.floor(rand.random() * 10000)) + 10
		for _ in range(c):
			# every vertex has max 100 edges
			e = math.floor(rand.random() * 102) + 1
			for __ in range(e):
				n = math.floor(rand.random() * (c)) + 1
				if n == _:
					continue
				self.add_pair(_, n, 1)

	def bfs(self, s = -2):
		d = deque()
		visited = []
		if s == -2:
			s = list(self.graph.keys())[0]
		d.append(s)
		visited.append(s)
		while d:
			s = d.popleft()
			if len(self.graph[s]) != 0:
				for __ in self.graph[s]:
					if visited.count(__[1]) < 1:
						d.append(__[1])
						visited.append(__[1])
		return visited
	def in_degree(self, u):
		count = 0
		for _ in self.graph:
			for __ in self.graph[_]:
				if __[1] == u:
					count += 1
		return count

	def out_degree(self, u):
		return len(self.graph[u])

	def topological_sort(self, s = -2):
		stack = []
		visited = []
		if s == -2:
			s = list(self.graph.keys())[0]
		stack.append(s)
		visited.append(s)
		ss = s
		sorted_nodes = []

		while True:
			# check if there is any non isolated nodes
			if len(self.graph[s]) != 0:
				ss = s
				for __ in self.graph[s]:
					if visited.count(__[1]) < 1:
						stack.append(__[1])
						visited.append(__[1])
						ss =__[1]
						break

			# check if all the children are visited
			if s == ss :
				sorted_nodes.append(stack.pop())
				if len(stack) != 0:
					s = stack[len(stack) - 1]
			else:
				s = ss

			# check if se have reached the starting point
			if len(stack) == 0:
				return sorted_nodes

	def cycle_nodes(self):
		stack = []
		visited = []
		s = list(self.graph.keys())[0]
		stack.append(s)
		visited.append(s)
		parent = -2
		indirect_parents = []
		ss = s
		anticipating_nodes = set()

		while True:
			# check if there is any non isolated nodes
			if len(self.graph[s]) != 0:
				ss = s
				for __ in self.graph[s]:
					if visited.count(__[1]) > 0 and __[1] != parent and indirect_parents.count(__[1]) > 0 and not on_the_way_back:
						l = len(stack) - 1
						while True and l >= 0:
							if stack[l] == __[1]:
								anticipating_nodes.add(__[1])
								break
							else:
								anticipating_nodes.add(stack[l])
								l -= 1
					if visited.count(__[1]) < 1:
						stack.append(__[1])
						visited.append(__[1])
						ss =__[1]
						break

			# check if all the children are visited
			if s == ss :
				stack.pop()
				on_the_way_back = True
				if len(stack) != 0:
					s = stack[len(stack) - 1]
			else:
				on_the_way_back = False
				indirect_parents.append(parent)
				parent = s
				s = ss

			# check if se have reached the starting point
			if len(stack) == 0:
				return list(anticipating_nodes)

	def has_cycle(self):
		stack = []
		visited = []
		s = list(self.graph.keys())[0]
		stack.append(s)
		visited.append(s)
		parent = -2
		indirect_parents = []
		ss = s
		anticipating_nodes = set()

		while True:
			# check if there is any non isolated nodes
			if len(self.graph[s]) != 0:
				ss = s
				for __ in self.graph[s]:
					if visited.count(__[1]) > 0 and __[1] != parent and indirect_parents.count(__[1]) > 0 and not on_the_way_back:
						l = len(stack) - 1
						while True and l >= 0:
							if stack[l] == __[1]:
								anticipating_nodes.add(__[1])
								break
							else:
								return True
								anticipating_nodes.add(stack[l])
								l -= 1
					if visited.count(__[1]) < 1:
						stack.append(__[1])
						visited.append(__[1])
						ss =__[1]
						break

			# check if all the children are visited
			if s == ss :
				stack.pop()
				on_the_way_back = True
				if len(stack) != 0:
					s = stack[len(stack) - 1]
			else:
				on_the_way_back = False
				indirect_parents.append(parent)
				parent = s
				s = ss

			# check if se have reached the starting point
			if len(stack) == 0:
				return False

class Graph:
	def __init__(self):
		self.graph = {}

	# adding vertices and edges
	# adding the weight is optional
	# handels repetition
	def add_pair(self, u, v, w = 1):
		# check if the u exists
		if self.graph.get(u):
			# if there already is a edge
			if self.graph[u].count([w,v]) == 0:
				self.graph[u].append([w, v])
		else:
			# if u does not exist
			self.graph[u] = [[w, v]]
		# add the other way
		if self.graph.get(v):
			# if there already is a edge
			if self.graph[v].count([w,u]) == 0:
				self.graph[v].append([w, u])
		else:
			# if u does not exist
			self.graph[v] = [[w, u]]
			
	# handels if the input does not exist
	def remove_pair(self, u, v):
		if self.graph.get(u):
			for _ in self.graph[u]:
				if _[1] == v:
					self.graph[u].remove(_)
		# the other way round
		if self.graph.get(v):
			for _ in self.graph[v]:
				if _[1] == u:
					self.graph[v].remove(_)

	# if no destination is meant the defaut value is -1
	def dfs(self, s = -2, d = -1):
		if s == d:
			return []
		stack = []
		visited = []
		if s == -2:
			s = list(self.graph.keys())[0]
		stack.append(s)
		visited.append(s)
		ss = s

		while True:
			# check if there is any non isolated nodes
			if len(self.graph[s]) != 0:
				ss = s
				for __ in self.graph[s]:
					if visited.count(__[1]) < 1:
						if __[1] == d:
							visited.append(d)
							return visited
						else:
							stack.append(__[1])
							visited.append(__[1])
							ss =__[1]
							break

			# check if all the children are visited
			if s == ss :
				stack.pop()
				if len(stack) != 0:
					s = stack[len(stack) - 1]
			else:
				s = ss

			# check if se have reached the starting point
			if len(stack) == 0:
				return visited	

	# c is the count of nodes you want and if you leave it or pass -1 to the funtion the count
	# will be random from 10 to 10000
	def fill_graph_randomly(self, c = -1):
		if c == -1:
			c = (math.floor(rand.random() * 10000)) + 10
		for _ in range(c):
			# every vertex has max 100 edges
			e = math.floor(rand.random() * 102) + 1
			for __ in range(e):
				n = math.floor(rand.random() * (c)) + 1
				if n == _:
					continue
				self.add_pair(_, n, 1)

	def bfs(self, s = -2):
		d = deque()
		visited = []
		if s == -2:
			s = list(self.graph.keys())[0]
		d.append(s)
		visited.append(s)
		while d:
			s = d.popleft()
			if len(self.graph[s]) != 0:
				for __ in self.graph[s]:
					if visited.count(__[1]) < 1:
						d.append(__[1])
						visited.append(__[1])
		return visited
	def degree(self, u):
		return len(self.graph[u])

	def cycle_nodes(self):
		stack = []
		visited = []
		s = list(self.graph.keys())[0]
		stack.append(s)
		visited.append(s)
		parent = -2
		indirect_parents = []
		ss = s
		anticipating_nodes = set()

		while True:
			# check if there is any non isolated nodes
			if len(self.graph[s]) != 0:
				ss = s
				for __ in self.graph[s]:
					if visited.count(__[1]) > 0 and __[1] != parent and indirect_parents.count(__[1]) > 0 and not on_the_way_back:
						l = len(stack) - 1
						while True and l >= 0:
							if stack[l] == __[1]:
								anticipating_nodes.add(__[1])
								break
							else:
								anticipating_nodes.add(stack[l])
								l -= 1
					if visited.count(__[1]) < 1:
						stack.append(__[1])
						visited.append(__[1])
						ss =__[1]
						break

			# check if all the children are visited
			if s == ss :
				stack.pop()
				on_the_way_back = True
				if len(stack) != 0:
					s = stack[len(stack) - 1]
			else:
				on_the_way_back = False
				indirect_parents.append(parent)
				parent = s
				s = ss

			# check if se have reached the starting point
			if len(stack) == 0:
				return list(anticipating_nodes)

	def has_cycle(self):
		stack = []
		visited = []
		s = list(self.graph.keys())[0]
		stack.append(s)
		visited.append(s)
		parent = -2
		indirect_parents = []
		ss = s
		anticipating_nodes = set()

		while True:
			# check if there is any non isolated nodes
			if len(self.graph[s]) != 0:
				ss = s
				for __ in self.graph[s]:
					if visited.count(__[1]) > 0 and __[1] != parent and indirect_parents.count(__[1]) > 0 and not on_the_way_back:
						l = len(stack) - 1
						while True and l >= 0:
							if stack[l] == __[1]:
								anticipating_nodes.add(__[1])
								break
							else:
								return True
								anticipating_nodes.add(stack[l])
								l -= 1
					if visited.count(__[1]) < 1:
						stack.append(__[1])
						visited.append(__[1])
						ss =__[1]
						break

			# check if all the children are visited
			if s == ss :
				stack.pop()
				on_the_way_back = True
				if len(stack) != 0:
					s = stack[len(stack) - 1]
			else:
				on_the_way_back = False
				indirect_parents.append(parent)
				parent = s
				s = ss

			# check if se have reached the starting point
			if len(stack) == 0:
				return False