"""simple docstring"""
# using dfs for finding eulerian path traversal







def 						__a   (	_SCREAMING_SNAKE_CASE		,     _SCREAMING_SNAKE_CASE		,     _SCREAMING_SNAKE_CASE		,     _SCREAMING_SNAKE_CASE=None   )	->Optional[int]:
  a__:     Optional[int]										=	(path or []) + [u]
  for v in graph[u]:
    if visited_edge[u][v] is False:
      a__							,    a__:     List[Any]										=	True, True
      a__:     Optional[Any]										=	dfs(_SCREAMING_SNAKE_CASE		,     _SCREAMING_SNAKE_CASE		,     _SCREAMING_SNAKE_CASE		,     _SCREAMING_SNAKE_CASE   )
  return path







def 						__a   (	_SCREAMING_SNAKE_CASE		,     _SCREAMING_SNAKE_CASE   )	->str:
  a__:     Optional[int]										=	0
  a__:     List[str]										=	-1
  for i in range(_SCREAMING_SNAKE_CASE   ):
    if i not in graph.keys():
      continue
    if len(graph[i]   ) % 2 == 1:
      odd_degree_nodes += 1
      a__:     Union[str, Any]										=	i
  if odd_degree_nodes == 0:
    return 1, odd_node
  if odd_degree_nodes == 2:
    return 2, odd_node
  return 3, odd_node







def 						__a   (	_SCREAMING_SNAKE_CASE		,     _SCREAMING_SNAKE_CASE   )	->str:
  a__:     Optional[int]										=	[[False for _ in range(max_node + 1   )] for _ in range(max_node + 1   )]
  a__							,    a__:     Union[str, Any]										=	check_circuit_or_path(_SCREAMING_SNAKE_CASE		,     _SCREAMING_SNAKE_CASE   )
  if check == 3:
    print('graph is not Eulerian'   )
    print('no path'   )
    return
  a__:     List[Any]										=	1
  if check == 2:
    a__:     Tuple										=	odd_node
    print('graph has a Euler path'   )
  if check == 1:
    print('graph has a Euler cycle'   )
  a__:     Union[str, Any]										=	dfs(_SCREAMING_SNAKE_CASE		,     _SCREAMING_SNAKE_CASE		,     _SCREAMING_SNAKE_CASE   )
  print(_SCREAMING_SNAKE_CASE   )







def 						__a   (	)	->List[Any]:
  a__:     Optional[Any]										=	{1: [2, 3, 4], 2: [1, 3], 3: [1, 2], 4: [1, 5], 5: [4]}
  a__:     int										=	{1: [2, 3, 4, 5], 2: [1, 3], 3: [1, 2], 4: [1, 5], 5: [1, 4]}
  a__:     List[Any]										=	{1: [2, 3, 4], 2: [1, 3, 4], 3: [1, 2], 4: [1, 2, 5], 5: [4]}
  a__:     Optional[int]										=	{1: [2, 3], 2: [1, 3], 3: [1, 2]}
  a__:     Optional[int]										=	{
      1: [],
      2: []
      # all degree is zero
  }
  a__:     Optional[Any]										=	10
  check_euler(_SCREAMING_SNAKE_CASE		,     _SCREAMING_SNAKE_CASE   )
  check_euler(_SCREAMING_SNAKE_CASE		,     _SCREAMING_SNAKE_CASE   )
  check_euler(_SCREAMING_SNAKE_CASE		,     _SCREAMING_SNAKE_CASE   )
  check_euler(_SCREAMING_SNAKE_CASE		,     _SCREAMING_SNAKE_CASE   )
  check_euler(_SCREAMING_SNAKE_CASE		,     _SCREAMING_SNAKE_CASE   )


if __name__ == "__main__":
 main()



"""simple docstring"""
class    __snake_case       :



      def __init__(						self    ,			lowercase    ,			lowercase=None    ,			lowercase=None)	->			List[str]:





        '''simple docstring'''





        a__:     Dict										=	data
        a__:     List[Any]										=	previous
        a__:     Any										=	next_node



      def __str__(						self)	->			str:





        '''simple docstring'''





        return f'{self.data}'



      def   lowerCamelCase_   (						self)	->			int:





        '''simple docstring'''





        return self.data



      def   lowerCamelCase_   (						self)	->			Union[str, Any]:





        '''simple docstring'''





        return self.next



      def   lowerCamelCase_   (						self)	->			Optional[Any]:





        '''simple docstring'''





        return self.previous







class    __snake_case       :



      def __init__(						self    ,			lowercase)	->			Dict:





        '''simple docstring'''





        a__:     List[Any]										=	head



      def __iter__(						self)	->			List[Any]:





        '''simple docstring'''





        return self



      def   lowerCamelCase_   (						self)	->			Optional[Any]:





        '''simple docstring'''





        if not self.current:
          raise StopIteration
        else:
          a__:     Dict										=	self.current.get_data()
          a__:     Optional[Any]										=	self.current.get_next()
          return value







class    __snake_case       :



      def __init__(						self)	->			Dict:





        '''simple docstring'''





        a__:     List[Any]										=	None  # First node in list
        a__:     Optional[int]										=	None  # Last node in list



      def __str__(						self)	->			Optional[Any]:





        '''simple docstring'''





        a__:     Dict										=	self.head
        a__:     Optional[Any]										=	[]
        while current is not None:
          nodes.append(current.get_data())
          a__:     str										=	current.get_next()
        return " ".join(str(lowercase) for node in nodes)



      def __contains__(						self    ,			lowercase)	->			Optional[int]:





        '''simple docstring'''





        a__:     Optional[int]										=	self.head
        while current:
          if current.get_data() == value:
            return True
          a__:     Dict										=	current.get_next()
        return False



      def __iter__(						self)	->			int:





        '''simple docstring'''





        return LinkedListIterator(self.head)



      def   lowerCamelCase_   (						self)	->			List[Any]:





        '''simple docstring'''





        if self.head:
          return self.head.get_data()
        return None



      def   lowerCamelCase_   (						self)	->			Optional[Any]:





        '''simple docstring'''





        if self.tail:
          return self.tail.get_data()
        return None



      def   lowerCamelCase_   (						self    ,			lowercase)	->			None:





        '''simple docstring'''





        if self.head is None:
          a__:     Optional[Any]										=	node
          a__:     Optional[Any]										=	node
        else:
          self.insert_before_node(self.head    ,			lowercase)



      def   lowerCamelCase_   (						self    ,			lowercase)	->			None:





        '''simple docstring'''





        if self.head is None:
          self.set_head(lowercase)
        else:
          self.insert_after_node(self.tail    ,			lowercase)



      def   lowerCamelCase_   (						self    ,			lowercase)	->			None:





        '''simple docstring'''





        a__:     Tuple										=	Node(lowercase)
        if self.head is None:
          self.set_head(lowercase)
        else:
          self.set_tail(lowercase)



      def   lowerCamelCase_   (						self    ,			lowercase    ,			lowercase)	->			None:





        '''simple docstring'''





        a__:     Union[str, Any]										=	node
        a__:     Optional[Any]										=	node.previous

        if node.get_previous() is None:
          a__:     Tuple										=	node_to_insert
        else:
          a__:     int										=	node_to_insert

        a__:     Optional[int]										=	node_to_insert



      def   lowerCamelCase_   (						self    ,			lowercase    ,			lowercase)	->			None:





        '''simple docstring'''





        a__:     Optional[int]										=	node
        a__:     Tuple										=	node.next

        if node.get_next() is None:
          a__:     Optional[int]										=	node_to_insert
        else:
          a__:     Any										=	node_to_insert

        a__:     str										=	node_to_insert



      def   lowerCamelCase_   (						self    ,			lowercase    ,			lowercase)	->			None:





        '''simple docstring'''





        a__:     Any										=	1
        a__:     Tuple										=	Node(lowercase)
        a__:     Tuple										=	self.head
        while node:
          if current_position == position:
            self.insert_before_node(lowercase    ,			lowercase)
            return
          current_position += 1
          a__:     List[Any]										=	node.next
        self.insert_after_node(self.tail    ,			lowercase)



      def   lowerCamelCase_   (						self    ,			lowercase)	->			Node:





        '''simple docstring'''





        a__:     Tuple										=	self.head
        while node:
          if node.get_data() == item:
            return node
          a__:     List[str]										=	node.get_next()
        raise Exception('Node not found')



      def   lowerCamelCase_   (						self    ,			lowercase)	->			Any:





        '''simple docstring'''





        if (node := self.get_node(lowercase)) is not None:
          if node == self.head:
            a__:     Any										=	self.head.get_next()

          if node == self.tail:
            a__:     List[Any]										=	self.tail.get_previous()

          self.remove_node_pointers(lowercase)



      @staticmethod
      def   lowerCamelCase_   (						lowercase)	->			None:





        '''simple docstring'''





        if node.get_next():
          a__:     Any										=	node.previous

        if node.get_previous():
          a__:     List[str]										=	node.next

        a__:     int										=	None
        a__:     Union[str, Any]										=	None



      def   lowerCamelCase_   (						self)	->			str:





        '''simple docstring'''





        return self.head is None







def 						__a   (	)	->None:
  pass


if __name__ == "__main__":
 import doctest

 doctest.testmod()


