



'''simple docstring'''






# using dfs for finding eulerian path traversal




def     snake_case_					(     lowerCAmelCase_       ,   lowerCAmelCase_       ,   lowerCAmelCase_       ,   lowerCAmelCase_=None		)->		List[str]:





       '''simple docstring'''


       _UpperCAmelCase  :						Any			     = (path or []) + [u]
       for v in graph[u]:
              if visited_edge[u][v] is False:
                     _UpperCAmelCase     ,_UpperCAmelCase  :						Tuple			     = True, True
                     _UpperCAmelCase  :						List[Any]			     = dfs(lowerCAmelCase_       ,   lowerCAmelCase_       ,   lowerCAmelCase_       ,   lowerCAmelCase_		)
       return path




def     snake_case_					(     lowerCAmelCase_       ,   lowerCAmelCase_		)->		Optional[int]:





       '''simple docstring'''


       _UpperCAmelCase  :						Optional[Any]			     = 0
       _UpperCAmelCase  :						Optional[int]			     = -1
       for i in range(lowerCAmelCase_		):
              if i not in graph.keys():
                     continue
              if len(graph[i]		) % 2 == 1:
                     odd_degree_nodes += 1
                     _UpperCAmelCase  :						Optional[int]			     = i
       if odd_degree_nodes == 0:
              return 1, odd_node
       if odd_degree_nodes == 2:
              return 2, odd_node
       return 3, odd_node




def     snake_case_					(     lowerCAmelCase_       ,   lowerCAmelCase_		)->		List[Any]:





       '''simple docstring'''


       _UpperCAmelCase  :						str			     = [[False for _ in range(max_node + 1		)] for _ in range(max_node + 1		)]
       _UpperCAmelCase     ,_UpperCAmelCase  :						int			     = check_circuit_or_path(lowerCAmelCase_       ,   lowerCAmelCase_		)
       if check == 3:
              print("""graph is not Eulerian"""		)
              print("""no path"""		)
              return
       _UpperCAmelCase  :						Dict			     = 1
       if check == 2:
              _UpperCAmelCase  :						Dict			     = odd_node
              print("""graph has a Euler path"""		)
       if check == 1:
              print("""graph has a Euler cycle"""		)
       _UpperCAmelCase  :						Dict			     = dfs(lowerCAmelCase_       ,   lowerCAmelCase_       ,   lowerCAmelCase_		)
       print(lowerCAmelCase_		)




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





       '''simple docstring'''


       _UpperCAmelCase  :						Any			     = {1: [2, 3, 4], 2: [1, 3], 3: [1, 2], 4: [1, 5], 5: [4]}
       _UpperCAmelCase  :						int			     = {1: [2, 3, 4, 5], 2: [1, 3], 3: [1, 2], 4: [1, 5], 5: [1, 4]}
       _UpperCAmelCase  :						Tuple			     = {1: [2, 3, 4], 2: [1, 3, 4], 3: [1, 2], 4: [1, 2, 5], 5: [4]}
       _UpperCAmelCase  :						List[Any]			     = {1: [2, 3], 2: [1, 3], 3: [1, 2]}
       _UpperCAmelCase  :						List[str]			     = {
           1: [],
           2: []
           # all degree is zero
       }
       _UpperCAmelCase  :						Union[str, Any]			     = 10
       check_euler(lowerCAmelCase_       ,   lowerCAmelCase_		)
       check_euler(lowerCAmelCase_       ,   lowerCAmelCase_		)
       check_euler(lowerCAmelCase_       ,   lowerCAmelCase_		)
       check_euler(lowerCAmelCase_       ,   lowerCAmelCase_		)
       check_euler(lowerCAmelCase_       ,   lowerCAmelCase_		)


if __name__ == "__main__":
   main()








'''simple docstring'''






class 							lowercase       :


 """simple docstring"""


 def __init__(							self							)		->   List[str]:
        _UpperCAmelCase  :						int			     = 0
        _UpperCAmelCase  :						Union[str, Any]			     = 0
        _UpperCAmelCase  :						Optional[int]			     = {}







 def _snake_case (							self  ,a_							)		->   Optional[Any]:
        if vertex not in self.adjacency:
               _UpperCAmelCase  :						int			     = {}
               self.num_vertices += 1







 def _snake_case (							self  ,a_  ,a_  ,a_							)		->   int:
        self.add_vertex(a_							)
        self.add_vertex(a_							)

        if head == tail:
               return

        _UpperCAmelCase  :						List[Any]			     = weight
        _UpperCAmelCase  :						Dict			     = weight







 def _snake_case (							self							)		->   Dict:
        _UpperCAmelCase  :						Optional[int]			     = self.get_edges()
        for edge in edges:
               _UpperCAmelCase     ,_UpperCAmelCase     ,_UpperCAmelCase  :						Dict			     = edge
               edges.remove((tail, head, weight)							)
        for i in range(len(a_							)							):
               _UpperCAmelCase  :						str			     = list(edges[i]							)

        edges.sort(key=lambda a_							: e[2]							)
        for i in range(len(a_							) - 1							):
               if edges[i][2] >= edges[i + 1][2]:
                      _UpperCAmelCase  :						Optional[Any]			     = edges[i][2] + 1
        for edge in edges:
               _UpperCAmelCase     ,_UpperCAmelCase     ,_UpperCAmelCase  :						Union[str, Any]			     = edge
               _UpperCAmelCase  :						str			     = weight
               _UpperCAmelCase  :						List[str]			     = weight







 def __str__(							self							)		->   Any:
        _UpperCAmelCase  :						List[Any]			     = """"""
        for tail in self.adjacency:
               for head in self.adjacency[tail]:
                      _UpperCAmelCase  :						List[str]			     = self.adjacency[head][tail]
                      string += f'''{head} -> {tail} == {weight}\n'''
        return string.rstrip("""\n"""							)







 def _snake_case (							self							)		->   str:
        _UpperCAmelCase  :						int			     = []
        for tail in self.adjacency:
               for head in self.adjacency[tail]:
                      output.append((tail, head, self.adjacency[head][tail])							)
        return output







 def _snake_case (							self							)		->   Optional[int]:
        return self.adjacency.keys()







 @staticmethod
 def _snake_case (							a_=None  ,a_=None							)		->   Tuple:
        _UpperCAmelCase  :						List[Any]			     = Graph()
        if vertices is None:
               _UpperCAmelCase  :						List[str]			     = []
        if edges is None:
               _UpperCAmelCase  :						Optional[Any]			     = []
        for vertex in vertices:
               g.add_vertex(a_							)
        for edge in edges:
               g.add_edge(*a_							)
        return g



 class 							lowercase       :


  """simple docstring"""


  def __init__(							self							)		->   int:
         _UpperCAmelCase  :						List[str]			     = {}
         _UpperCAmelCase  :						int			     = {}







  def __len__(							self							)		->   Tuple:
         return len(self.parent							)







  def _snake_case (							self  ,a_							)		->   str:
         if item in self.parent:
                return self.find(a_							)

         _UpperCAmelCase  :						Optional[Any]			     = item
         _UpperCAmelCase  :						List[Any]			     = 0
         return item







  def _snake_case (							self  ,a_							)		->   List[str]:
         if item not in self.parent:
                return self.make_set(a_							)
         if item != self.parent[item]:
                _UpperCAmelCase  :						List[Any]			     = self.find(self.parent[item]							)
         return self.parent[item]







  def _snake_case (							self  ,a_  ,a_							)		->   Union[str, Any]:
         _UpperCAmelCase  :						Any			     = self.find(a_							)
         _UpperCAmelCase  :						List[str]			     = self.find(a_							)

         if roota == roota:
                return roota

         if self.rank[roota] > self.rank[roota]:
                _UpperCAmelCase  :						Any			     = roota
                return roota

         if self.rank[roota] < self.rank[roota]:
                _UpperCAmelCase  :						Any			     = roota
                return roota

         if self.rank[roota] == self.rank[roota]:
                self.rank[roota] += 1
                _UpperCAmelCase  :						List[str]			     = roota
                return roota
         return None







 @staticmethod
 def _snake_case (							a_							)		->   List[Any]:
        _UpperCAmelCase  :						int			     = graph.num_vertices

        _UpperCAmelCase  :						int			     = Graph.UnionFind()
        _UpperCAmelCase  :						Optional[int]			     = []
        while num_components > 1:
               _UpperCAmelCase  :						int			     = {}
               for vertex in graph.get_vertices():
                      _UpperCAmelCase  :						Union[str, Any]			     = -1

               _UpperCAmelCase  :						Tuple			     = graph.get_edges()
               for edge in edges:
                      _UpperCAmelCase     ,_UpperCAmelCase     ,_UpperCAmelCase  :						str			     = edge
                      edges.remove((tail, head, weight)							)
               for edge in edges:
                      _UpperCAmelCase     ,_UpperCAmelCase     ,_UpperCAmelCase  :						Optional[Any]			     = edge
                      _UpperCAmelCase  :						Any			     = union_find.find(a_							)
                      _UpperCAmelCase  :						Any			     = union_find.find(a_							)
                      if seta != seta:
                             if cheap_edge[seta] == -1 or cheap_edge[seta][2] > weight:
                                    _UpperCAmelCase  :						Tuple			     = [head, tail, weight]

                             if cheap_edge[seta] == -1 or cheap_edge[seta][2] > weight:
                                    _UpperCAmelCase  :						List[str]			     = [head, tail, weight]
               for vertex in cheap_edge:
                      if cheap_edge[vertex] != -1:
                             _UpperCAmelCase     ,_UpperCAmelCase     ,_UpperCAmelCase  :						str			     = cheap_edge[vertex]
                             if union_find.find(a_							) != union_find.find(a_							):
                                    union_find.union(a_  ,a_							)
                                    mst_edges.append(cheap_edge[vertex]							)
                                    _UpperCAmelCase  :						Tuple			     = num_components - 1
        _UpperCAmelCase  :						Optional[int]			     = Graph.build(edges=a_							)
        return mst



